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We consider the interpolation inequality with respect to the regularity index s ∈ R in homogeneous Besov spaces B ˙ p , q s ( R n ) . By choosing a general summability index 1 ≤ q ≤ ∞ and estimating carefully, we reveal a precise representation of the constant appearing in the interpolation inequality. As an application of the refined interpolation inequality, we show a generalization of the Gagliardo–Nirenberg inequality in homogeneous Besov spaces given by Wadade (2009). [ABSTRACT FROM AUTHOR]

Let n≥2n\ge 2 and Ω⊂Rn\Omega \subset {{\mathbb{R}}}^{n} be a bounded nontangentially accessible domain. In this article, the authors investigate (weighted) global gradient estimates for Dirichlet boundary value problems of second-order elliptic equations of divergence form with an elliptic symmetric part and a BMO antisymmetric part in Ω\Omega . More precisely, for any given p∈(2,∞)p\in \left(2,\infty ), the authors prove that a weak reverse Hölder inequality with exponent pp implies the global W1,p{W}^{1,p} estimate and the global weighted W1,q{W}^{1,q} estimate, with q∈[2,p]q\in \left[2,p] and some Muckenhoupt weights, of solutions to Dirichlet boundary value problems. As applications, the authors establish some global gradient estimates for solutions to Dirichlet boundary value problems of second-order elliptic equations of divergence form with small BMO{\rm{BMO}} symmetric part and small BMO{\rm{BMO}} antisymmetric part, respectively, on bounded Lipschitz domains, quasi-convex domains, Reifenberg flat domains, C1{C}^{1} domains, or (semi-)convex domains, in weighted Lebesgue spaces. Furthermore, as further applications, the authors obtain the global gradient estimate, respectively, in (weighted) Lorentz spaces, (Lorentz–)Morrey spaces, (Musielak–)Orlicz spaces, and variable Lebesgue spaces. Even on global gradient estimates in Lebesgue spaces, the results obtained in this article improve the known results via weakening the assumption on the coefficient matrix.

In this paper we present a bilinear estimate for commutators on Triebel–Lizorkin spaces with variable smoothness and integrability, and under no vanishing assumptions on the divergence of vector fields. [ABSTRACT FROM AUTHOR]

Questions concerning quantitative and asymptotic properties of the elliptic measure corresponding to a uniformly elliptic divergence form operator have been the focus of recent studies. In this setting we show that the elliptic measure of an operator with coefficients satisfying a vanishing Carleson condition in the upper half space is an asymptotically optimal A ∞ weight. In particular, for such operators the logarithm of the elliptic kernel is in the space of (locally) vanishing mean oscillation. To achieve this, we prove local, quantitative estimates on a quantity (introduced by Fefferman, Kenig and Pipher) that controls the A ∞ constant. Our work uses recent results obtained by David, Li and Mayboroda. These quantitative estimates may offer a new framework to approach similar problems. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the parabolic off-diagonal Muckenhoupt Ap,q+ weights. We reveal the properties of these weights, and our main results characterize them through weak-type and strong-type weighted norm inequalities for the fractional forward-in-time maximal operator. Moreover, our results cover some well-known results. [ABSTRACT FROM AUTHOR]

We study quasilinear elliptic equations of the type - Δ p ⁢ u = σ ⁢ u q + μ in ℝ n in the case 0 < q < p - 1 , where μ and σ are nonnegative measurable functions, or locally finite measures, and Δ p ⁢ u = div ⁡ ( | ∇ ⁡ u | p - 2 ⁢ ∇ ⁡ u) is the p-Laplacian. Similar equations with more general local and nonlocal operators in place of Δ p are treated as well. We obtain existence criteria and global bilateral pointwise estimates for all positive solutions u: u ⁢ (x) ≈ (퐖 p ⁢ σ ⁢ (x)) p - q p - q - 1 + 퐊 p , q ⁢ σ ⁢ (x) + 퐖 p ⁢ μ ⁢ (x) , x ∈ ℝ n , where 퐖 p and 퐊 p , q are, respectively, the Wolff potential and the intrinsic Wolff potential, with the constants of equivalence depending only on p, q, and n. The contributions of μ and σ in these pointwise estimates are totally separated, which is a new phenomenon even when p = 2 . [ABSTRACT FROM AUTHOR]

Let Ω ⊂ ℝ n + 1 , n ≥ 2 , be a 1-sided non-tangentially accessible domain (aka uniform domain), that is, Ω satisfies the interior Corkscrew and Harnack chain conditions, which are respectively scale-invariant/quantitative versions of openness and path-connectedness. Let us assume also that Ω satisfies the so-called capacity density condition, a quantitative version of the fact that all boundary points are Wiener regular. Consider L 0 ⁢ u = - div ⁢ (A 0 ⁢ ∇ ⁡ u) , L ⁢ u = - div ⁢ (A ⁢ ∇ ⁡ u) , two real (non-necessarily symmetric) uniformly elliptic operators in Ω, and write ω L 0 , ω L for the respective associated elliptic measures. The goal of this program is to find sufficient conditions guaranteeing that ω L satisfies an A ∞ -condition or a RH q -condition with respect to ω L 0 . In this paper we establish that if the discrepancy of the two matrices satisfies a natural Carleson measure condition with respect to ω L 0 , then ω L ∈ A ∞ ⁢ (ω L 0) . Additionally, we can prove that ω L ∈ RH q ⁢ (ω L 0) for some specific q ∈ (1 , ∞) , by assuming that such Carleson condition holds with a sufficiently small constant. This "small constant" case extends previous work of Fefferman–Kenig–Pipher and Milakis–Pipher together with the last author of the present paper who considered symmetric operators in Lipschitz and bounded chord-arc domains, respectively. Here we go beyond those settings, our domains satisfy a capacity density condition which is much weaker than the existence of exterior Corkscrew balls. Moreover, their boundaries need not be Ahlfors regular and the restriction of the n-dimensional Hausdorff measure to the boundary could be even locally infinite. The "large constant" case, that is, the one on which we just assume that the discrepancy of the two matrices satisfies a Carleson measure condition, is new even in the case of nice domains (such as the unit ball, the upper-half space, or non-tangentially accessible domains) and in the case of symmetric operators. We emphasize that our results hold in the absence of a nice surface measure: all the analysis is done with the underlying measure ω L 0 , which behaves well in the scenarios we are considering. When particularized to the setting of Lipschitz, chord-arc, or 1-sided chord-arc domains, our methods allow us to immediately recover a number of existing perturbation results as well as extend some of them. [ABSTRACT FROM AUTHOR]

We prove a quadratic sparse domination result for general non-integral square functions S. That is, for p 0 ∈ [ 1 , 2) and q 0 ∈ (2 , ∞ ] , we prove an estimate of the form where q 0 ∗ is the Hölder conjugate of q 0 / 2 , M is the underlying doubling space and S is a sparse collection of cubes on M. Our result will cover both square functions associated with divergence form elliptic operators and those associated with the Laplace–Beltrami operator. This sparse domination allows us to derive optimal norm estimates in the weighted space L p (w) . [ABSTRACT FROM AUTHOR]

We develop a new real-variable method for weighted L p estimates. The method is applied to the study of weighted W 1 , 2 estimates in Lipschitz domains for weak solutions of second-order elliptic systems in divergence form with bounded measurable coefficients. It produces a necessary and sufficient condition, which depends on the weight function, for the weighted W 1 , 2 estimate to hold in a fixed Lipschitz domain with a given weight. Using this condition, for elliptic systems in Lipschitz domains with rapidly oscillating, periodic and VMO coefficients, we reduce the problem of weighted estimates to the case of constant coefficients. [ABSTRACT FROM AUTHOR]

A generalized incompressable magnetohydrodynamics system is considered in this paper. Furthermore, results of global well-posednenss are established with the aid of Littlewood–Paley decomposition and Fourier localization method in mentioned system with small initial condition in the variable exponent Fourier–Besov–Morrey spaces. Moreover, the Gevrey class regularity of the solution is also achieved in this paper. [ABSTRACT FROM AUTHOR]

Let n ≥ 3 , Ω be a strongly Lipschitz domain of R n , and p (·) : R n → (0 , 1 ] a variable exponent function satisfying the globally log-Hölder continuous condition. Assume that L Ω : = - Δ + V is a Schrödinger operator on L 2 (Ω) with the Dirichlet boundary condition, where Δ denotes the Laplace operator and the nonnegative potential V belongs to the reverse Hölder class R H q 0 (R n) for some q 0 ∈ (n / 2 , ∞ ] . In this article, the authors first introduce the variable Hardy space H L Ω p (·) (Ω) associated with L Ω on Ω , via the Lusin area function associated with L Ω , and the "geometrical" variable Hardy space H L R n , r p (·) (Ω) , via the variable Hardy space H L R n p (·) (R n) associated with the Schrödinger operator L R n : = - Δ + V on R n , and then prove that H L Ω p (·) (Ω) = H L R n , r p (·) (Ω) with equivalent quasi-norms. As an application, the authors show that, when Ω is a bounded, simply connected, and semiconvex domain of R n and the nonnegative potential V belongs to the reverse Hölder class R H q 0 (R n) for some q 0 ∈ (max { n / 2 , 2 } , ∞ ] , the operators V L Ω - 1 and ∇ 2 L Ω - 1 are bounded from H L R n , r p (·) (Ω) to the variable Lebesgue space L p (·) (Ω) , or to itself. As a corollary, the second-order regularity for the inhomogeneous Dirichlet problems of the corresponding Schrödinger equations in the scale of variable Hardy spaces H L R n , r p (·) (Ω) is obtained. [ABSTRACT FROM AUTHOR]

The ripplet transform is a refinement of the ordinary curvelet transform based on the non-parabolic scaling law for resolving two-dimensional singularities. However, the ripplet transform offers flexibility only in the angular window which is not feasible for optimizing the concentration of the ripplet spectrum. Therefore, there arises a fundamental question “whether it is possible to induce flexibility in both the radial and angular windows of the ordinary ripplet transform”. The aim of this study is to address this issue by constructing a new family of ripplet waveforms based on the linear canonical transform having certain extra degrees of freedom for resolving higher dimensional singularities. To begin with, we examine the fundamental aspects of the novel ripplet transform including the formulation of reconstruction and Rayleigh’s energy formulae. In the sequel, we obtain a Heisenberg-type uncertainty inequality associated with the novel ripplet transform. Towards the culmination, we introduce and study a new class of localization operators associated with the novel ripplet transform. [ABSTRACT FROM AUTHOR]

Let n ≥ 2 , Ω ⊂ R n be a bounded non-tangentially accessible domain, and p (·) : R n → (0 , ∞) a variable exponent function satisfying 0 < p - ≤ p + < ∞ , where p - : = ess inf x ∈ R n p (x) and p + : = ess sup x ∈ R n p (x) . Assume that L D is a second order divergence form elliptic operator having real-valued, bounded, and measurable coefficients on L 2 (Ω) with the Dirichlet boundary condition. The main aim of this article is threefold. First, the authors establish the molecular characterization for the variable Hardy space H L D p (·) (Ω) associated with L D . In particular, when L D is non-negative and self-adjoint, the authors further obtain the atomic and the maximal function characterization of H L D p (·) (Ω) . Secondly, the authors introduce the “geometrical” variable Hardy space H r p (·) (Ω) and its local version h r p (·) (Ω) by restricting any element of the variable Hardy space H p (·) (R n) and its local version h p (·) (R n) , respectively, to Ω , and then show that, when p - ∈ (n n + θ , 1 ] , H p (·) (Ω) = H r p (·) (Ω) = H L D p (·) (Ω) = h r p (·) (Ω) with equivalent quasi-norms, where H p (·) (Ω) denotes the variable Hardy space on Ω and θ ∈ (0 , 1 ] is the critical index of Hölder continuity for the heat kernels { p t , L D } t > 0 generated by L D . Thirdly, as applications, the authors prove the boundedness of the Riesz transform ∇ L D - 1 / 2 on the variable Lebesgue space L p (·) (Ω) when 1 < p - ≤ p + ≤ 2 , and from H L D p (·) (Ω) to L p (·) (Ω) when 0 < p - ≤ p + ≤ 1 , or to H p (·) (Ω) when n n + 1 < p - ≤ p + ≤ 1 . Meanwhile, the authors also show the boundedness of the fractional integral L D - β from L p (·) (Ω) to L q (·) (Ω) , when 1 < p (·) < q (·) < ∞ , and from H L D p (·) (Ω) to L q (·) (Ω) , when 0 < p (·) ≤ 1 < q (·) < ∞ , or to H L D q (·) (Ω) , when 0 < p (·) < q (·) ≤ 1 , where β ∈ (0 , n 2) and 1 p (·) - 1 q (·) = 2 β n . As a corollary, global gradient estimates, in both L p (·) (Ω) when 1 < p - ≤ p + ≤ 2 and H p (·) (Ω) when n n + 1 < p - ≤ p + ≤ 1 , for the inhomogeneous Dirichlet problem of L D on Ω are obtained. Finally, a div-curl lemma in the space H r p (·) (Ω) is established. [ABSTRACT FROM AUTHOR]

The well-posedness of stochastic Navier–Stokes equations with various noises is a hot topic in the area of stochastic partial differential equations. Recently, the consideration of stochastic Navier–Stokes equations involving fractional Laplacian has received more and more attention. Due to the scaling-invariant property of the fractional stochastic equations concerned, it is natural and also very important to study the well-posedness of stochastic fractional Navier–Stokes equations in the associated critical Fourier–Besov spaces. In this paper, we are concerned with the three-dimensional stochastic fractional Navier–Stokes equation driven by multiplicative noise. We aim to establish the well-posedness of solutions of the concerned equation. To this end, by utilising the Fourier localisation technique, we first establish the local existence and uniqueness of the solutions in the critical Fourier–Besov space B ˙ p , r 4 - 2 α - 3 p . Then, under the condition that the initial date is sufficiently small, we show the global existence of the solutions in the probabilistic sense. [ABSTRACT FROM AUTHOR]

We prove weighted norm inequalities with Muckenhoupt’s A p -weights, for a wide class of oscillatory integral operators. As a consequence, one also obtains the boundedness of commutators of the aforementioned operators with functions in BMO. [ABSTRACT FROM AUTHOR]

We establish new local and global estimates for evolutionary partial differential equations in classical Banach and quasi-Banach spaces that appear most frequently in the theory of partial differential equations. More specifically, we obtain optimal (local in time) estimates for the solution to the Cauchy problem for variable-coefficient evolutionary partial differential equations. The estimates are achieved by introducing the notions of Schrödinger and general oscillatory integral operators with inhomogeneous phase functions and prove sharp local and global regularity results for these in Besov–Lipschitz and Triebel–Lizorkin spaces.

In the past decades, we learnt that uniform rectifiability is often a right candidate to go past Lipschitz boundaries in boundary value problems. If Ω is an open domain in R n with mild topological conditions, we can even characterize the n - 1 dimensional uniformly rectifiability of the boundary ∂ Ω by the A ∞ -absolute continuity of the harmonic measure on ∂ Ω with respect to the surface measure. In low dimension ( d < n - 1 ), David and Mayboroda tackled one direction of the above characterization, i.e. proved that if Γ is a d-dimensional uniformly rectifiable set, then the harmonic measure (associated to an suitable degenerate elliptic operator) on Γ is A ∞ -absolutely continuous with respect to the d-dimensional Hausdorff measure. In the present article, we use a completely new approach to give an alternative and significantly shorter proof of David and Mayboroda's result. [ABSTRACT FROM AUTHOR]

We explore the connection between k-broad Fourier restriction estimates and sharp regularity L p - L q local smoothing estimates for the solutions of the wave equation in R n × R for all n ≥ 3 via a Bourgain–Guth broad-narrow analysis. An interesting feature is that local smoothing estimates for e i t - Δ are not invariant under Lorentz rescaling. [ABSTRACT FROM AUTHOR]

In this paper, several direct and inverse theorems in terms of the best approximations of functions and moduli of smoothness are proved concerning the approximation of functions from the space L 2 (α , β) by partial sums of Jacobi-Dunkl series. For this purpose, we use the generalized Jacobi-Dunkl translation operator which was defined by Vinogradov. [ABSTRACT FROM AUTHOR]

This paper surveys recent progress on the analysis of the □ b operator on quadric submanifolds of C n × C m . We focus our discussion on the (relative) fundamental solution to □ b on quadric submanifolds of arbitrary codimension. We summarize known results regarding □ b -invariance of mappings, necessary and sufficient conditions for solvability and hypoellipticity for □ b , and we describe the L p regularity of the complex Green operator on quadrics with nonvanishing Levi form. We discuss the ramifications of these results for many examples of quadrics of codimension 1 and 2. [ABSTRACT FROM AUTHOR]

It was observed recently in work of Bez, Buschenhenke, Cowling, Flock and the first author, that the euclidean Brascamp–Lieb inequality satisfies a natural and useful Fourier duality property. The purpose of this paper is to establish an appropriate discrete analogue of this. Our main result identifies the Brascamp–Lieb constants on (finitely-generated) discrete abelian groups with Brascamp–Lieb constants on their (Pontryagin) duals. As will become apparent, the natural setting for this duality principle is that of locally compact abelian groups, and this raises basic questions about Brascamp–Lieb constants formulated in this generality. [ABSTRACT FROM AUTHOR]

This article is concerned with the incompressible, infinite-depth water wave equation in two space dimensions, with gravity and constant vorticity but with no surface tension. We consider this problem expressed in position-velocity potential holomorphic coordinates, and prove local well-posedness for large data, as well as cubic lifespan bounds for small-data solutions.

This paper establishes pointwise estimates up to boundary for the gradient of weak solutions to a class of very singular quasilinear elliptic equations with mixed data

Let w be a Muckenhoupt A2(ℝn) weight and Ω a bounded Reifenberg flat domain in ℝn. Assume that p (·):Ω → (1, ∞) is a variable exponent satisfying the log-Hölder continuous condition. In this article, the authors investigate the weighted W1, p (·)(Ω, w)-regularity of the weak solutions of second order degenerate elliptic equations in divergence form with Dirichlet boundary condition, under the assumption that the degenerate coefficients belong to weighted BMO spaces with small norms.

Motivated by water-wave problems, in this paper we consider a class ofnonlinear dispersive PDEs in 2D with cubic nonlinearities, whose dispersionrelations are radial and have vanishing Guassian curvature on a circle. Forsuch a model we identify certain null structures for the cubic nonlinearity,which suffice in order to guarantee global scattering solutions for the smalldata problem. Our null structures in the power-type nonlinearity are weak, andonly eliminate the worst nonlinear interaction. Such null structures arisenaturally in some water-wave problems.

This article is concerned with the incompressible, irrotational infinite depth water wave equation in two space dimensions, without gravity but with surface tension. We consider this problem expressed in position–velocity potential holomorphic coordinates, and prove that small data solutions have at least cubic lifespan while small localized data leads to global solutions.

In this article we consider irrotational gravity water waves with finite bottom. Our goal is two-fold. First, we represent the equations in holomorphic coordinates and discuss the local well-posedness of the problem in this context. Second, we consider the small data problem and establish cubic lifespan bounds for the solutions. Our results are uniform in the infinite depth limit, and match our earlier infinite depth result in Hunter et al. (Commun Math Phys, 2014).

Let k ∈ { 1 , 2 , 3 … } , d ∈ { 2 k + 1 , 2 k + 2 , … } and V ∈ R H σ ∩ G k , d (- Δ) k with σ ≥ d / 2 + k . Here R H σ is a reverse Holder class and G k , d (- Δ) k is a Gaussian class associated with (- Δ) k . Consider the parabolic Schrodinger operator L = ∂ t + (- Δ) k + V k . We show that for certain k, α and V the Riesz transforms ∇ x 2 k L - 1 and V k α L - α are bounded on L p (R + d + 1) for suitable values of p ∈ (1 , ∞) . [ABSTRACT FROM AUTHOR]

In this paper, we get an Lp boundedness of Fourier integral operators with rough amplitude a ∈ L ∞ S ϱ m , and phase φ ∈ L ∞ Φ 2 for 1 ≤ p ≤ + ∞ . This is an improvement of the corresponding results in Dos Santos Ferreira and Staubach (Mem. Amer. Math. Soc. 229, 1074, 2014). [ABSTRACT FROM AUTHOR]

Let X be a space of homogeneous type with the doubling order n. Let L be a nonnegative self-adjoint operator on L 2 (X) and suppose that the kernel of e - t L satisfies a Gaussian upper bound. This paper shows that for 0 < p ≤ 1 and s = n (1 / p - 1 / 2) , ‖ (I + L) - s e itL f ‖ H L p (X) ≲ (1 + | t |) s ‖ f ‖ H L p (X) for all t ∈ R , where H L p (X) is the Hardy space associated to L. This recovers the classical results in the particular case when L = - Δ and extends a number of known results. [ABSTRACT FROM AUTHOR]

We prove that for antisymmetric vector field Ω with small L 2 -norm there exists a gauge A ∈ L ∞ ∩ W ˙ 1 / 2 , 2 (R 1 , G L (N)) such that div 1 2 (A Ω - d 1 2 A) = 0. This extends a celebrated theorem by Rivière to the nonlocal case and provides conservation laws for a class of nonlocal equations with antisymmetric potentials, as well as stability under weak convergence. [ABSTRACT FROM AUTHOR]

We study L p improving estimates and continuity properties of maximal operators for generalized spherical means. Once these features are obtained, they are applied to get sparse bounds on lacunary and full generalized spherical averages. [ABSTRACT FROM AUTHOR]

Let (X, d, μ) be a metric measure space with non-negative Ricci curvature. This paper is concerned with the boundary behavior of harmonic function on the (open) upper half-space X × ℝ+. We derive that a function f of bounded mean oscillation (BMO) is the trace of harmonic function u(x, t) on X × ℝ+, u(x, 0) = f(x), whenever u satisfies the following Carleson measure condition sup x B , r B ∫ 0 r B ∫ B (x B , r B) | t ∇ u (x , t) | 2 d μ (x) d t t ≤ C < ∞ , where ∇ = (∇x,∂t) denotes the total gradient and B(xB, rB) denotes the (open) ball centered at xB with radius rB. Conversely, the above condition characterizes all the harmonic functions whose traces are in BMO space. [ABSTRACT FROM AUTHOR]

This article is concerned with the infinite depth water wave equation in two space dimensions. We consider this problem expressed in position-velocity potential holomorphic coordinates. Viewing this problem as a quasilinear dispersive equation, we establish two results: (i) local well-posedness in Sobolev spaces, and (ii) almost global solutions for small localized data. Neither of these results are new; they have been recently obtained by Alazard–Burq–Zuily (Invent Math 198:71–163, 2014), respectively by Wu (Invent Math 177:45–135, 2009) using different coordinates and methods. Instead our goal is to improve the understanding of this problem by providing a single setting for both problems, by proving sharper versions of the above results, as well as presenting new, simpler proofs. This article is self contained.

Through conformal map, isoperimetric inequalities are equivalent to the Hardy–Littlewood–Sobolev (HLS) inequalities involved with the Poisson-type kernel on the upper half space. From the analytical point of view, we want to know whether there exists a reverse analogue for the Poisson-type kernel. In this work, we give an affirmative answer to this question. We first establish the reverse Stein–Weiss inequality with the Poisson-type kernel, finding that the range of index 𝑝,q′q^{\prime} appearing in the reverse inequality lies in the interval (0,1)(0,1), which is perfectly consistent with the feature of the index for the classical reverse HLS and Stein–Weiss inequalities. Then we give the existence and asymptotic behaviors of the extremal functions of this inequality. Furthermore, for the reverse HLS inequalities involving the Poisson-type kernel, we establish the regularity for the positive solutions to the corresponding Euler–Lagrange system and give the sufficient and necessary conditions of the existence of their solutions. Finally, in the conformal invariant index, we classify the extremal functions of the latter reverse inequality and compute the sharp constant. Our methods are based on the reversed version of the Hardy inequality in high dimension, Riesz rearrangement inequality and moving spheres.

We consider the operator L = - div (A ∇) , where A is an n × n matrix of real coefficients and satisfies the ellipticity condition, with n ≥ 2 . We assume that the coefficients of the symmetric part of A are in L ∞ (R n) , and those of the anti-symmetric part of A only belong to the space B M O (R n) . We create a complete narrative of the L p theory for the square root of L and show that it satisfies the L p estimates L f L p ≲ ∇ f L p for 1 < p < ∞ , and ∇ f L p ≲ L f L p for 1 < p < 2 + ϵ for some ϵ > 0 depending on the ellipticity constant and the BMO semi-norm of the coefficients. Moreover, we prove the L p estimates for some vertical square functions associated to e - t L . In another article of the authors, these results are used to establish the solvability of the Dirichlet problem for elliptic equation div (A (x) ∇ u) = 0 in the upper half-space (x , t) ∈ R + n + 1 with the boundary data in L p (R n , d x) for some p ∈ (1 , ∞) . [ABSTRACT FROM AUTHOR]

Let L be a Schrödinger operator of the form L = - Δ + V acting on L 2 (R n) where the non-negative potential V belongs to the reverse Hölder class RH q for some q ≥ (n + 1) / 2 . Let CMO L (R n) denote the function space of vanishing mean oscillation associated to L . In this article, we will show that a function f of CMO L (R n) is the trace of the solution to L u = - u tt + L u = 0 , u (x , 0) = f (x) , if and only if, u satisfies a Carleson condition sup B : balls C u , B : = sup B (x B , r B) : balls r B - n ∫ 0 r B ∫ B (x B , r B) | t ∇ u (x , t) | 2 dx dt t < ∞ , and lim a → 0 sup B : r B ≤ a C u , B = lim a → ∞ sup B : r B ≥ a C u , B = lim a → ∞ sup B : B ⊆ B (0 , a) c C u , B = 0. This continues the lines of the previous characterizations by Duong et al. (J Funct Anal 266(4):2053–2085, 2014) and Jiang and Li () for the BMO L spaces, which were founded by Fabes et al. (Indiana Univ Math J 25:159–170, 1976) for the classical BMO space. For this purpose, we will prove two new characterizations of the CMO L (R n) space, in terms of mean oscillation and the theory of tent spaces, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we first prove a general symmetrization principle for the Hardy type inequality with non-radial weights of the form A x x 1 P 1 ... x N P N (Theorem 1.1). Using this symmetrization principle for Hardy's inequalities, we can establish the improved L p -Hardy inequalities with such non-radial monomial weights (Theorem 1.2). Second, we introduce the notion of p-Bessel pairs and give applications to L p -Hardy identities with non-radial monomial weights (Theorem 1.3) and Hardy inequalities (see Theorem 1.4) and their virtual extremals (see Remark 1.2). (See Theorem 1.5 for the special case p = 2 where we derive L 2 -Hardy identities and inequalities with monomial weights which have not been studied in the literature). In the special case when P = (0 , ... , 0) , they imply the L p -Hardy identities and Hardy inequalities with remainder terms on any finite balls and the entire space R N (Theorem 1.6), while in the special case P = (0 , ... , 0 , α) , α > 0 , our results provide the L p -Hardy identities and Hardy inequalities on half balls and the half spaces (Theorem 1.7). By taking special examples of p-Bessel pairs, we establish some particular Hardy's identities and weighted Sobolev inequalities which are of independent interest. We also establish the optimal Hardy inequalities with monomial weights and explicit forms of extremal functions. (See Corollaries 1.1, 1.2, 1.3, 1.4, 1.5.) Our above results sharpened earlier results in the literature even in the case of L 2 Hardy inequalities. Finally, we establish the sharp constants and optimal functions of the L p -Hardy–Sobolev inequalities with monomial weights (Theorem 1.8). [ABSTRACT FROM AUTHOR]

We prove the local smoothing estimate for general Fourier integral operators with phase function of the form ϕ (x , t , ξ) = x · ξ + t q (ξ) , with q ∈ C ∞ (R 2 \ { 0 }) , homogeneous of degree one, and amplitude functions in the symbol class of order m ≤ 0 . The result is global in the space variable, and also improves our previous work in this direction (Manna et al (in: Georgiev et al., Advances in harmonic analysis and partial differential equations, Trends in Mathematics. Birkhäuser, Cham, pp. 1–35, 2020)). The approach involves a reduction to operators with amplitude function depending only on the covariable, and a new estimate for square function based on angular decomposition. [ABSTRACT FROM AUTHOR]

The paper is mainly concerned with the interconnection of the boundary behaviour of the solutions of the exterior Dirichlet and Neumann problems of harmonic analysis for the unit disk in R 2 and the unit ball in R 3 with the corresponding behaviour of the associated ergodic inverse problems for the entire space. The basis is the theory of semigroups of linear operators mapping a Banach space X into itself. The classical one-parameter theory for semigroups applies in the present particular applications, actually for X = L 2 π 2 in case of the unit disk, and X = L 2 (S) in the three dimensional setting, S being the unit sphere in R 3 . Another tool is a Drazin-like inverse operator B for the infinitesimal generator A of a semigroup that arises naturally in ergodic theory. This operator B is a closed, not necessarily bounded, operator. It was introduced in a paper with Butzer and Westphal (Indiana Univ Math J 20:1163–1174, 1970/1971) and extended to a generalized setting with Butzer and Koliha (J Oper Theory 62:297–326, 2009). [ABSTRACT FROM AUTHOR]

In this paper, the authors obtain the Dunkl analogy of classical Lp Hardy inequality for p > N + 2γ with sharp constant ( p − N − 2 γ p) p , where 2γ is the degree of weight function associated with Dunkl operators, and Lp Hardy inequalities with distant function in some G-invariant domains. Moreover they prove two Hardy-Rellich type inequalities for Dunkl operators. [ABSTRACT FROM AUTHOR]

Let b ∈ L loc 1 (R n) and α ∈ (0 , 1). In the present paper, we are concerned on commutators with fractional differentiations which are defined by [ b , T α ] f (x) = p. v. ∫ R n Ω (x - y) | x - y | n + α (b (x) - b (y)) f (y) d y. This kind of commutators [ b , T α ] appear in the parabolic boundary value problems. In this paper, we obtain the necessary and sufficient conditions on the function b to guarantee that [ b , T α ] is a bounded operator on L p (w) for 1 < p < ∞ and w ∈ A p. For the sufficiency, we study the quantitative weighted bounds for the commutator with fractional differentiation with rough kernels. Namely, for b ∈ I α (B M O) and 1 ≤ r ′ < p < ∞ , ‖ [ b , T α ] f ‖ L p (w) ≤ C ‖ Ω ‖ L r ‖ D α b ‖ BMO [ w ] A p / r ′ max 1 , 1 p - r ′ [ w ] A p / r ′ 1 r ′ max 1 , r ′ p - r ′ ‖ f ‖ L p (w) , it is the best known quantitative result of this operator. For the necessity, if [ b , T α ] is bounded on L p (w) for some 1 < p < ∞ and w ∈ A p , then b ∈ L i p α. Finally, we apply our general theory to the Hilbert and Riesz transforms, and obtain a characterization of [ b , D α R j ] and [ b , D α H ] on L p (w) for 1 < p < ∞ and w ∈ A p by b ∈ I α (B M O) . [ABSTRACT FROM AUTHOR]

In nice environments, such as Lipschitz or chord-arc domains, it is well-known that the solvability of the Dirichlet problem for an elliptic operator in L p , for some finite p, is equivalent to the fact that the associated elliptic measure belongs to the Muckenhoupt class A ∞ . In turn, any of these conditions occurs if and only if the gradient of every bounded null solution satisfies a Carleson measure estimate. This has been recently extended to much rougher settings such as those of 1-sided chord-arc domains, that is, sets which are quantitatively open and connected with a boundary which is Ahlfors–David regular. In this paper, we work in the same environment and consider a qualitative analog of the latter equivalence showing that one can characterize the absolute continuity of the surface measure with respect to the elliptic measure in terms of the finiteness almost everywhere of the truncated conical square function for any bounded null solution. As a consequence of our main result particularized to the Laplace operator and some previous results, we show that the boundary of the domain is rectifiable if and only if the truncated conical square function is finite almost everywhere for any bounded harmonic function. In addition, we obtain that for two given elliptic operators L 1 and L 2 , the absolute continuity of the surface measure with respect to the elliptic measure of L 1 is equivalent to the same property for L 2 provided the disagreement of the coefficients satisfy some quadratic estimate in truncated cones for almost everywhere vertex. Finally, for the case on which L 2 is either the transpose of L 1 or its symmetric part we show the equivalence of the corresponding absolute continuity upon assuming that the antisymmetric part of the coefficients has some controlled oscillation in truncated cones for almost every vertex. [ABSTRACT FROM AUTHOR]

In this paper, we establish some Hardy inequalities for the p-Laplace operator associated with Robin boundary conditions in general domains. Moreover, by using the new Hardy inequality, we derive the classical Hardy inequality with the weight of mean distance function. [ABSTRACT FROM AUTHOR]