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In this paper, we investigate the removable problem of isolated singularities for fourth order elliptic equations with nonstandard growth. Based on Moser’s iteration technique, the sufficient condition is found for removability of singularity for the fourth order elliptic equations in the framework of variable exponent Sobolev spaces.

In this paper, we provide a constructive proof of $\mathbf {H}^{1}(\mathbb {R}^{n})$ factorization in terms of multilinear Calderon–Zygmund operators in Morrey spaces. As a direct application, we obtain a characterization of functions in $\text {BMO}(\mathbb {R}^{n})$ via commutators of multilinear Calderon–Zygmund operators. Furthermore, we prove a Morrey compactness characterization of [b, T]l, the commutator in the l-th entry.

We consider positive classical solutions of $$ \left\{ \begin{array}{rcl} -{\Delta} u + \lambda a(x) \cdot \nabla u &=& u^{p} \qquad\text{in} {\Omega}, \\ u&=& 0 \qquad \quad \text{on} {\partial {\Omega}}, \end{array}\right. $$ where p > 1, a is a smooth divergence free vector field and λ > 0 is a large parameter. Under certain assumptions on a(x) and (or) assumptions on the existence of first integrals of a(x) we show there is a subsequence of smooth positive solutions which converge to a nonzero first integral of a(x) as $ \lambda \rightarrow \infty $ .

For harmonic functions v on the disc, it has been known for a long time that non-tangential boundedness a.e.is equivalent to finiteness a.e. of the integral of the area function of v (Lusin area theorem). This result also hold for functions that are non-tangentially bounded only in a measurable subset of the boundary, and has been extended to rank-one hyperbolic spaces, and also to infinite trees (homogeneous or not). No equivalent of the Lusin area theorem is known on higher rank symmetric spaces, with the exception of the degenerate higher rank case given by the cartesian product of rank-one hyperbolic spaces. Indeed, for products of two discs, an area theorem for jointly harmonic functions was proved by M.P. and P. Malliavin, who introduced a new area function; non-tangential boundedness a.e. is a sufficient condition, but not necessary, for the finiteness of this area integral. Their result was later extended to general products of rank-one hyperbolic spaces by Korányi and Putz. Here we prove an area theorem for jointly harmonic functions on the product of a finite number of infinite homogeneous trees; for the sake of simplicity, we give the proofs for the product of two trees. This could be the first step to an area theorem for Bruhat–Tits affine buildings, thereby shedding light on the higher rank continuous set-up.

In this paper, we study the Sobolev extension property of Lp-quasidisks which are the generalizations of classical quasidisks. After that, we also find some applications of this property.

We prove a Korovkin-type theorem for sequences of positive linear operators on C([0, 1]). This theorem allows us to discuss some general results concerning positive linear operators. In particular, we give simple and efficient convergence criteria for the iterates of positive linear operators preserving the affine functions. By way of examples, we consider the Bernstein, the Meyer-Konig and Zeller, the q-Bernstein operators and their iterates.

Motivated by the equivalent definition of a continuous operator between Banach spaces in terms of weakly null nets, we introduce unbounded continuous operators by replacing weak convergence with the unbounded absolutely weak convergence (uaw-convergence) in the definition of a continuous operator between Banach lattices. We characterize order continuous Banach lattices and reflexive Banach lattices in terms of these spaces of operators. Moreover, motivated by characterizing of a reflexive Banach lattice in terms of unbounded absolutely weakly Cauchy sequences, we consider pre-unbounded continuous operators between Banach lattices which maps uaw-Cauchy sequences to weakly (uaw- or norm) convergent sequences. This allows us to characterize KB-spaces and reflexive spaces in terms of these operators, too. Furthermore, we consider the unbounded Banach–Saks property as an unbounded version of the weak Banach–Saks property. There are many relations between spaces possessing the unbounded Banach–Saks property with spaces fulfilled by different types of the known Banach–Saks property. In particular, we characterize order continuous Banach lattices in terms of these relations, as well.

This paper introduces a ‘new’ class of operators. An operator T from a Banach lattice E into a Banach space Y will be called an LW-compact if T carries L-weakly compact sets of E into relatively compact sets of Y. Various characterizations of LW-compact operators are given. Also, we study the domination problem in this class of operators.

We consider a nonlinear Dirichlet problem driven by the (p, q)-Laplacian and with a Caratheodory reaction f(z, x) which is ( $$p-1$$ )-superlinear in $$x\in {\mathbb {R}}$$ but without satisfying the Ambrosetti–Rabinowitz condition. Using variational tools and critical groups, we prove a multiplicity theorem producing three nontrivial smooth solutions all with sign information (one positive, one negative and one nodal).

This paper deals with (1, 1; r)-convexity of sequences. First, we prove several results on the sets of (1, 1; r)-convex sequences for various values of r. Then we show that the Cesaro matrix does not preserve (1, 1; r)-convexity of sequences for any $$r\ge 2$$ .

In this paper, we introduce polynomial versions of the weak Dunford–Pettis property and the weak Dunford–Pettis $$^{*}$$ property for Banach lattices. By using Fremlin projective Banach lattice tensor products, we obtain several characterizations of the weak Dunford–Pettis property and the weak Dunford–Pettis $$^{*}$$ property in terms of regular homogeneous polynomials on Banach lattices.

We prove new inequalities for the spectral radius, essential spectral radius, operator norm, measure of noncompactness and numerical radius of Hadamard weighted geometric mean of (infinite or finite) nonnegative matrices that define positive operators on Banach sequence spaces. Some of these inequalities complement the known inequalities and some of them refine known inequalities. Several inequalities appear to be new even in the finite dimensional case.

In this note, we establish a property of the higher-order generalized Studniarski epiderivative. By virtue of the property, we demonstrate that conditions (6) and (12) of Theorem 4.3 in Anh (Positivity 22:1371–1385, 2018) are incompatible. We provide a modification of Theorem 4.3 in Anh (Positivity 22:1371–1385, 2018). An example is given to illustrate the modified result.

The aim of this note is to consider different notions for the Banach–Saks property in locally solid vector lattices as an extension for the known concepts of the Banach–Saks property in Banach lattices. We investigate relations between them; in particular, we shall characterize spaces in which, these notions agree.

In this paper, we study the boundedness of multilinear commutators of Calderon–Zygmund operators $$T_{\mathbf {b}}$$ on generalized variable exponent Morrey spaces $$M^{p(\cdot ), \varphi }$$ . Let $$\mathbf {b}=(b_1,\ldots ,b_m)$$ and $$b_i \in BMO$$ for $$i=1,\ldots ,m$$ . Then the sufficient conditions on the pair $$(\varphi _1,\varphi _2)$$ , which ensure the boundedness of the operator $$T_{\mathbf {b}}$$ from $$M^{p(\cdot ), \varphi _1}$$ to $$M^{p(\cdot ), \varphi _2}$$ , are found.

The classical multidimensional version of Fatou’s lemma (Schmeidler in Proc Am Math Soc 24:300–306, 1970) originally obtained for unconditional expectations and the standard non-negative cone in a finite-dimensional linear space is extended to conditional expectations and general closed pointed cones.

We investigate the 3rd term of the spectral heat content for killed subordinate and subordinate killed Brownian motions on a bounded open interval D = (a,b) in a real line when the underlying subordinators are stable subordinators with index α ∈ (1, 2) or α = 1. We prove that in the 3rd term of the spectral heat content, one can observe the length b − a of the interval D.

We consider the PDE - Δ p ⁢ u = ρ -\Delta_{p}u=\rho , where 𝜌 is a signed Borel measure on R n \mathbb{R}^{n} . For each p > n p>n , we characterize solutions as extremals of a generalized Morrey inequality determined by 𝜌.

In this paper we extend the construction of Grand and Small Lebesgue spaces for the case of general Banach function spaces on finite measure space. We call these spaces the grand and the small $$X^p$$ spaces. We prove results on several fundamental properties of these spaces, namely, duality, rearrangement invariant and the other properties that are transferred from the original space X to the corresponding grand and small spaces. In particular, on duality, we show that the generalized associate space of the small $$X^p$$ space with respect to the Banach function space X is the corresponding grand $$X^p$$ space.

In this paper, we give some sufficient and necessary conditions for discrete cosine operator functions on solid Banach function spaces to be chaotic or topologically transitive.

In this paper, we get an Lp boundedness of Fourier integral operators with rough amplitude $a\in L^{\infty } S^{m}_{\varrho },~$ and phase $\varphi \in L^{\infty }{\Phi }^{2}$ for $1\leq p\leq +\infty $ . This is an improvement of the corresponding results in Dos Santos Ferreira and Staubach (Mem. Amer. Math. Soc. 229, 1074, 2014).

We deal with the existence and multiplicity of solutions for a non-local fractional system involving combined critical non-linearities. Some existence results are obtained by splitting the Nehari manifold and applying the Mountain–Pass theorem of Ambrosetti and Rabinowitz.

2-Local derivations on real matrix algebras over unital semi-prime Banach algebras are considered. Using the real analogue of the result that any 2-local derivation on the algebra $$M_{2^n}(A)$$ M 2 n ( A ) ($$n\ge 2$$ n ≥ 2 ) is a derivation, it is shown that any 2-local derivation on real AW$$^*$$ ∗ -algebra for which the enveloping algebra is (complex) AW*-algebra, is a derivation, where A is a unital semi-prime Banach algebra with the inner derivation property.

We deal with existence and uniqueness of positive solutions of an elliptic boundary value problem modeled by $$ \left \{\begin {array}{ll} \displaystyle -{\Delta }_{p} u= \frac {f}{u^{\gamma }} + g u^{q} & \text { in } {\Omega }, \\ u = 0 & \text {on } \partial {\Omega }, \end {array}\right . $$ − Δ p u = f u γ + g u q in Ω , u = 0 on ∂ Ω , where Ω is an open bounded subset of $\mathbb {R}^{N}$ ℝ N where Ω is an open bounded subset of $\mathbb {R}^{N}$ ℝ N , Δpu := ÷(|∇u|p− 2∇u) is the usual p-Laplacian operator, γ ≥ 0 and 0 ≤ q ≤ p − 1; f and g are nonnegative functions belonging to suitable Lebesgue spaces.

The necessary and sufficient conditions for both the Kadec–Klee property as well as the Kadec–Klee property with respect to the coordinatewise convergence in Orlicz–Lorentz sequence spaces equipped with the Orlicz norm and generated by arbitrary Orlicz functions as well as any non-increasing weight sequences are given. Moreover, for their subspaces of elements with an order continuous norm the full characterization of the Kadec–Klee property with respect to the coordinatewise convergence is presented. Some tools useful in the proofs of the main results are also provided.

In this paper, we prove a reverse Holder inequality for the eigenfunction of the Dirichlet problem on domains of a compact Riemannian manifold with the integral Ricci curvature condition. We also prove an isoperimetric inequality for the torsional rigidity of such domains. These results extend some recent work of Gamara et al. (Open Math. 13(1), 557–570, 2015) and Colladay et al. (J. Geom. Anal. 28(4), 3906–3927, 2018) from the pointwise lower Ricci curvature bound to the integral Ricci curvature condition. We also extend the results from Laplacian to p-Laplacian.

We study the local regularity of solutions f to the integro-differential equation $$ Af=g \quad \text{in } U $$ A f = g in U for open sets $U \subseteq \mathbb {R}^{d}$ U ⊆ ℝ d , where A is the infinitesimal generator of a Lévy process (Xt)t≥ 0. Under the assumption that the transition density of (Xt)t≥ 0 satisfies a certain gradient estimate, we establish interior Schauder estimates for both pointwise and weak solutions f. Our results apply for a wide class of Lévy generators, including generators of stable Lévy processes and subordinated Brownian motions.

We introduce generalized local and global Herz spaces where all their characteristics are variable. As one of the main results we show that variable Morrey type spaces and complementary variable Morrey type spaces are included into the scale of these generalized variable Herz spaces. We also prove the boundedness of a class of sublinear operators in generalized variable Herz spaces with application to variable Morrey type spaces and their complementary spaces, based on the mentioned inclusion. United Arab Emirates University, Al Ain, United Arab Emirates [G00002994]; Russian Foundation for Basic ResearchRussian Foundation for Basic Research (RFBR) [19-01-00223, 20-51-46003]; TUBITAKTurkiye Bilimsel ve Teknolojik Arastirma Kurumu (TUBITAK) info:eu-repo/semantics/publishedVersion

Assume that X is a reflexive Banach space with an unconditional basis. In this paper, we show that the Banach space K(X) of compact operators on X has Pelczynski’s property V.

We discuss positive Miyadera–Voigt type perturbations for bi-continuous semigroups on $$\mathrm {AL}$$ -spaces with an additional locally convex topology generated by additive seminorms. The main example of such spaces is the space of bounded Borel measures (on a Polish space).

In this paper, we study Sobolev type inequalities for fractional maximal functions in the half space. We also discuss Sobolev type inequalities for Green potentials in the half space.

In this paper, we define the $$L_p$$ -mixed affine surface area probability measure and obtain the log-Minkowski inequalities for nonsymmetric convex bodies of $$L_p$$ -mixed affine surface area. Besides, we also establish the functional inequalities for $$L_p$$ -mixed affine surface area, which are more general form than the log-Minkowski inequalities of $$L_p$$ -mixed affine surface area.

Under mild assumptions, we establish a Liouville theorem for the "Laplace" equation $Au=0$ associated with the infinitesimal generator $A$ of a L\'evy process: If $u$ is a weak solution to $Au=0$ which is at most of (suitable) polynomial growth, then $u$ is a polynomial. As a by-product, we obtain new regularity estimates for semigroups associated with L\'evy processes., Comment: Proof of Proposition 2.2 has been substantially simplified

A necessary and sufficient condition is given to ensure the boundedness of Fourier–Haar multiplier operators from $$L^1 ([0, 1], X)$$ to $$L^1 ([0, 1], Y)$$ , where X is an arbitrary finite dimensional Banach space and Y is an arbitrary Banach space. The Fourier–Haar multiplier sequences come not from $${\mathbb {R}}$$ , as in the classical case, but from the space of bounded operators from the Banach space X to the Banach space Y. Moreover, it is shown that this condition characterises the finite dimensionality of the Banach space X.

Firstly we consider a finite dimensional Markov semigroup generated by Dunkl Laplacian with drift terms. For this semigroup we prove gradient bounds involving a symmetrised carre-du-champ operator, and we show that for small coefficients this semigroup has a unique invariant measure which satisfies ergodicity properties. We then extend this analysis to an infinite dimensional model on $(\mathbb {R}^{N})^{\mathbb {Z}^{d}}$ , consisting of interacting finite dimensional models. We construct an associated Markov semigroup for this model using gradient bounds, and finally we study the existence of invariant measures and ergodicity properties.

We consider parabolic equations of the form $$ u_{t}-\text{div} \left( |\nabla u|^{p-2}\nabla u+ a(x,t)|\nabla u|^{q-2}\nabla u\right)= 0, a(x,t)\geq 0. $$ In the range $\frac {2n}{n+1}

We investigate the different notions of solutions to the double-phase equation - div ⁡ ( | D ⁢ u | p - 2 ⁢ D ⁢ u + a ⁢ ( x ) ⁢ | D ⁢ u | q - 2 ⁢ D ⁢ u ) = 0 , -{\operatorname{div}(\lvert Du\rvert^{p-2}Du+a(x)\lvert Du\rvert^{q-2}Du)}=0, which is characterized by the fact that both ellipticity and growth switch between two different types of polynomial according to the position. We introduce the 𝒜 H ⁢ ( ⋅ ) {\mathcal{A}_{H(\,{\cdot}\,)}} -harmonic functions of nonlinear potential theory and then show that 𝒜 H ⁢ ( ⋅ ) {\mathcal{A}_{H(\,{\cdot}\,)}} -harmonic functions coincide with the distributional and viscosity solutions, respectively. This implies that the distributional and viscosity solutions are exactly the same.

In this paper, we prove existence, uniqueness, and local regularity of the solution to the Sobolev–Dirichlet problem for quasilinear elliptic equations in the generalized Orlicz–Sobolev spaces on domains, not necessarily bounded, of $${\mathbb {R}}^N$$ . Our approach is based on solving the obstacle problem and using the Harnack inequality.

A classical theorem of Bocher says a positive harmonic function on a punctured domain $\mathcal {O}\setminus p$ in Euclidean space can be written as the sum of a constant multiple of the Green function with pole at p and a function harmonic on all of $\mathcal {O}$ . Here we establish such a result in the setting of functions on a Riemannian manifold with rather rough metric tensor.

Area minimizing hypersurfaces and, more generally, almost minimizing hypersurfaces frequently occur in geometry, dynamics and physics. A central problem is that a general (almost) minimizing hypersurfaceHcontains a complicated singular set Σ. Nevertheless, we manage to develop a detailed potential theory onH∖Σ applicable to large classes of linear elliptic second order operators. We even get a fine control over their analysis near Σ. This is Part 2 of this two parts work.

For possibly discontinuous functions including, for instance, Sobolev functions, we present new Blaschke-Privaloff-type criteria for superharmonicity and harmonicity. This opens the way for introduction of a substantial generalization of the Laplace operator. These potential-theoretic considerations lead to a new kind of non-absolutely convergent integral where the integrand may be a highly oscillating pointwise function or even a distribution-valued function. In turn, this integral gives a precise meaning to some generalized Poisson equations with a wild right hand side.

In this paper we establish Sobolev type compact embedding theorems for Hormander classes of pseudodifferential operators $OpS^{-\alpha }_{1,\delta }$ on localizable Hardy space. Our work include new optimal boundedness results. As application, we obtain compact embeddings for compactly supported distributions with respect to the space variables in the nonhomogeneous localizable Hardy-Sobolev spaces.

In this paper we prove that the $$\ell $$ -group tensor product of two Archimedean f-rings is again an f-ring. We will use this result to characterize multiplicative $$\ell $$ -bimorphisms between unital f-rings.

The purpose of this paper is to provide necessary and sufficient conditions of the boundedness for singular integrals on the local Hardy space and its dual. Particularly the singular integrals considered in this paper include the pseudo-differential operators $T_{\sigma }f(x)=\int \limits \sigma (x \xi )e^{2\pi ix\xi }\hat {f}(\xi )d\xi $ with $\sigma \in S_{1 0}^{0}$ . As a consequence our results give another proof of the boundedness of the pseudo-differential operators on the local Hardy space (Goldberg Duke Math. J. 46(1) 27–42 1979).

The Ando–Choi–Effros lifting theorem provides conditions under which a bounded linear mapping taking values in a quotient space can be lifted through the quotient map. We prove two versions of said theorem for regular maps between Banach lattices. Our conditions mirror the classical ones, but additionally taking into account the order structure.

In this paper, we investigate conditionally completely positive definite kernels in the setting of Hilbert $$C^*$$ -modules. At first, we present a correspondence between conditionally completely positive definite kernels and completely positive definite kernels. Utilizing this relation, we give the Kolmogorov decomposition for a certain conditionally completely positive definite kernel. We present a characterization of conditionally completely positive definite kernels majorized by a kernel under some mild conditions. Finally, as an application of this decomposition, we find a sufficient condition for the existence of an extreme point of a convex set consisting of some special kernels.

We study the size of the set of points where the α-divided difference of a function in the Holder class Λα is bounded below by a fixed positive constant. Our results are obtained from their discrete analogues which can be stated in the language of dyadic martingales. Our main technical result in this setting is a sharp estimate of the Hausdorff measure of the set of points where a dyadic martingale with bounded increments has maximal growth.

As a variant of metric subregularity, pseudo metric subregularity is studied via general limit critical sets using the techniques of variational analysis. In terms of limit critical sets, we provide some sufficient conditions for the validity of pseudo/Holder metric subregularity. Usually, the property of pseudo metric subregularity is not stable under small smooth perturbation. We provide a characterization for pseudo metric subregularity to be stable under small $$C^{1,p}$$ smooth perturbation. In particular, some existing results on metric subregularity are extended to pseudo metric subregularity. Finally, we consider the pseudo weak sharp minimizer of a proper lower semicontinuous function and its relation with pseudo metric subregularity of the corresponding subdifferential mapping.

Here, we show the equivalent conditions for integrability and being almost everywhere zero of trace-class operator-valued functions with respect to some spectral measure. Furthermore, we investigate the relation between the convergence of these functions and the one of their integrals.

In this paper, under several hypotheses and using a dual gap functional, weak sharp solutions are studied for a multidimensional variational-type inequality governed by $$(\rho , \mathbf {b}, \mathbf {d})$$ -convex multiple integral functional. Moreover, a relation between the minimum principle sufficiency property and weak sharpness of solutions for the considered multidimensional variational-type inequality is established.