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We obtain a new characterization of the higher Sobolev space $W^{m,p}(\mathbb R^{n})$ W m , p ( ℝ n ) , $m\in \mathbb N$ m ∈ ℕ and $p\in (1, +\infty )$ p ∈ ( 1 , + ∞ ) and of the space BVm, the space of functions of higher order bounded variation. The characterizations are in term of BMO-type seminorms. The results unify and substantially extend previous results in Fusco et al. (ESAIM Control Optim. Calc Var., 24(2), 835–847 2018) and Farroni et al. (J. Funct. Anal., 278(9), 108451 2020).

The aim of this paper is to prove a reverse Holder inequality for nonnegative adjoint solutions for elliptic operator in non divergence form in \begin{document}$\mathbb R^3$\end{document} . As an application we generalize a Theorem due to Sirazhudinov and Zhikov [ 24 ] and, under suitable assumptions, we characterize the \begin{document}$G$\end{document} -limit of a sequence of elliptic operator.The operator \begin{document}$N$\end{document} \begin{document}$N[v] = \sum\limits_{i,j = 1}^3 {\frac{{{\partial ^2}({a_{ij}}v)}}{{\partial {x_i}\partial {x_j}}}} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( 1 \right)$ \end{document} arises naturally as the formal adjoint of the operator in "non divergence form" \begin{document}$L[u] = \sum\limits_{i,j = 1}^3 {{a_{i,j}}} (x)\frac{{{\partial ^2}u}}{{\partial {x_i}\partial {x_j}}} = Tr(A{D^2}u).\;\;\;\;\;\;\;\;\;\;\;\;\;\left( 2 \right)$ \end{document} The reason to study the solutions of the adjoint operator is that they are not only important for the solvability of \begin{document}$Lu = f$\end{document} but for the properties of the Green's function for \begin{document}$L$\end{document} . There is a long literature in this context, see for example Syogren [ 22 ], Bauman [ 2 ], Fabes and Stroock [ 12 ], Fabes, Garofalo, Marĺn-Malave, and Salsa [ 11 ], Escauriaza and Kenig [ 10 ], and Escauriaza [ 9 ].

In this paper we consider a linear elliptic equation in divergence form $$\begin{aligned} \sum _{i,j}D_j(a_{ij}(x)D_i u )=0 \quad \hbox {in } \Omega . \end{aligned}$$ ∑ i , j D j ( a ij ( x ) D i u ) = 0 in Ω . Assuming the coefficients $$a_{ij}$$ a ij in $$W^{1,n}(\Omega )$$ W 1 , n ( Ω ) with a modulus of continuity satisfying a certain Dini-type continuity condition, we prove that any very weak solution $$u\in L^{n'}_\mathrm{loc}(\Omega )$$ u ∈ L loc n ′ ( Ω ) of (0.1) is actually a weak solution in $$W^{1,2}_\mathrm{loc}(\Omega )$$ W loc 1 , 2 ( Ω ) .

Given a Banach space E with a supremum-type norm induced by a collection of operators, we prove that E is a dual space and provide an atomic decomposition of its predual. We apply this result, and some results obtained previously by one of the authors, to the function space B introduced recently by Bourgain, Brezis, and Mironescu. This yields an atomic decomposition of the predual B ⁎ , the biduality result that B 0 ⁎ = B ⁎ and B ⁎ ⁎ = B , and a formula for the distance from an element f ∈ B to B 0 .

A homeomorphism U of the unit disk D ⊂ R 2 , U = ( u 1 , u 2 ) : D ⟶ onto D is a quasiharmonic map, if u i ∈ W l o c 1 , 1 , i = 1 , 2 are solutions to the system (0.1) { div B ( y ) ∇ u 1 = 0 a.e. in D div B ( y ) ∇ u 2 = 0 a.e. in D for a symmetric degenerate elliptic conductivity B = B ( y ) i.e. (0.2) ∣ ξ ∣ 2 H ( y ) ≤ 〈 B ( y ) ξ , ξ 〉 ≤ H ( y ) ∣ ξ ∣ 2 a.e. in y ∈ D ∀ ξ ∈ R 2 where H : D ⟶ onto [ 1 , ∞ [ is measurable. A sufficient condition that the Sobolev homeomorphism U ∈ W l o c 1 , 1 is a quasiharmonic map is that U − 1 ∈ W loc 1 , 1 . This condition is not necessary because we construct a quasiharmonic map U such that U − 1 ∈ BV ∖ W loc 1 , 1 (see Theorem 1.1 ).

We consider pairs of non reflexive Banach spaces $$(E_0, E)$$ such that $$E_0$$ is defined in terms of a little-o condition and E is defined by the corresponding big-O condition. Under suitable assumptions on the pair $$(E_0, E)$$ there exists a reflexive and separable Banach space X (in which E is continuously embedded and dense) naturally associated to E which characterizes quantitatively weak compactness of bounded linear operators $$\begin{aligned} T: E_0 \rightarrow Z \end{aligned}$$ where Z is an arbitrary Banach space. Pairs include (VMO, BMO), where BMO is the space of John-Nirenberg, $$(B_0, B)$$ where B is a recently introduced space by Bourgain-Brezis-Mironescu ([6]) and some Orlicz pairs $$(L^{\psi }_0, L^{\psi })$$ where $$L^{\psi }_0$$ is the closure of $$L^\infty $$ in the Orlicz space $$L^{\psi }$$ , Marcinkiewicz pairs $$(L^{q, \infty }_0, L^{q, \infty })$$ where $$L^{q, \infty }_0$$ is the closure of $$L^\infty $$ in the Marcinkiewicz weak– $$L^q$$ denoted by $$L^{q, \infty }$$ . More generally, Banach function spaces are considered. The main results are duality formulas of the type 1 $$\begin{aligned} E_0^{**}&\simeq E\qquad \text {isometrically}\end{aligned}$$ 2 $$\begin{aligned}E^{*}&\simeq E_0^*\oplus _1 E_0^\perp\end{aligned}$$ and distance formulas.

Let Ω ⊆ ℝ2 be a domain, let X be a rearrangement invariant space and let f ∈ W1 X (Ω, ℝ2) be a homeomorphism between Ω and f(Ω). Then there exists a sequence of diffeomorphisms fk converging to f in the space W1X (Ω, ℝ2).

In the recent literature certain BMO-type seminorms provide characterizations of Sobolev functions. In the same order of ideas, we obtain the norm of the gradient of a function in L p ( Ω ) , where Ω ⊂ R n , n > 1 and p > 1 , as limit of BMO-type seminorms involving families of pairwise disjoint sets with all orientations, the sets being not necessarily cubes or tessellation cells. An analogous result is obtained when rotations are not allowed.

The purpose of this paper is to present the relation between certain BMO-type seminorms and the total variation of SBV functions. Following some ideas of a recent paper by L. Ambrosio and G.E. Comi, we give a representation formula of the total variation of SBV functions which does not make use of the distributional derivatives. We consider an anisotropic variant of the BMO-type seminorm introduced in 2015 in a paper by J. Bourgain, H. Brezis and P. Mironescu, by using, instead of cubes, covering families made by translations of a given open bounded set with Lipschitz boundary.

In this paper we introduce the class of the inner p-quasiconformal mappings, that are homeomorphisms f : D ⟶ o n t o D , f ∈ W loc 1 , 1 ( D ; D ) , where D ⊂ R 2 is the unit disk, such that there exists a constant K p ≥ 0 for which the following distortion inequality | D f ( x ) | p ≤ K p | J f ( x ) | p − 1 a.e. x ∈ D is satisfied. The study of such mappings is motivated by the fact that their inverses satisfy the distortion inequality introduced in [11] . Here we give a characterization of them so that their components solve a suitable uniformly elliptic p-harmonic system. Moreover, for mappings satisfying the previous distortion inequality with K p = K p , f ( x ) not necessarily constant, we identify the homeomorphism f whose p-distortion function K p , f ( x ) is minimal in L 1 norm.

This note is concerned with the Poisson equation in the unit disk of the complex plane. Our setting is in the Sobolev space W1;Þ with exponent 1 < p < infty. Such a setting with p eq2 is referred to as beyond the natural domain of definition. The novelty lies in the use of a singular integral called Beurling Transform.

For a planar domain Ω, we study the Dirichlet problem for the quasilinear elliptic equation $$ - divA\left( {x,\nabla v} \right) = f$$ when f belongs to the Zygmund space \(L{\left( {\log L} \right)^{\frac{1}{2}}}{\left( {\log \log \log L} \right)^{\frac{\beta }{2}}}\left( \Omega \right),\beta \ge 0.\). We prove that there exists a unique solution \(v \in W_0^{1,2}\left( \Omega \right)\) with \(\left| {\nabla v} \right| \in {L^2}{\left( {\log \log \log L} \right)^\beta }\left( \Omega \right)\).

We prove an embedding theorem for functions of bounded variation. We give an application to the study of Poisson integral on the disk.

Buckley (1993) [3] proved the linear dependence ‖ M ‖ L 2 ( R n , w ) ≤ c ( n ) A of the L 2 ( R n , w ) -norm for the Hardy–Littlewood maximal operator M on the classical A 2 -constant A = A 2 ( w ) = sup Q ⨏ Q w ⨏ Q w − 1 where the supremum is taken over all cubes with sides parallel to the axes. We prove in the case n = 1 that, for p 0 = 1 + A − 1 A p ≤ 2 , the dependence on the constant A is precisely preserved ‖ M ‖ L p ( R , w ) ≤ c ( p ) [ A 1 − p ( 2 − p ) A ] 1 p − 1 and it is impossible to decrease the value of p 0 . Similar exact continuation theorems hold for the L 2 -norm inequalities of weighted maximal operators.

Given a planar domain Ω, we study the Dirichlet problem $$\textstyle\begin{cases} {-}\operatorname {div}A(x, \nabla v)= f & \mbox{in } \Omega,\\ v=0 & \mbox{on } \partial\Omega, \end{cases} $$ where the higher-order term is a quasilinear elliptic operator, and f belongs to the Zygmund space $L (\log L)^{\delta} (\log\log\log L)^{\frac{\beta}{2}}(\Omega)$ with $\beta\geq0$ and $\delta\geq \frac{1}{2}$ . We prove that the gradient of the variational solution $v \in W^{1,2}_{0}(\Omega)$ belongs to the space $L^{2} (\log L)^{2\delta-1} (\log \log\log L)^{\beta}(\Omega)$ .

Let Ω ⊂ R n be an open set. We show that for a continuous mapping f ∈ W 1 , n − 1 ( Ω , R n ) with J f ∈ L 1 ( Ω ) the validity of the Lusin (N) condition implies that the distributional Jacobian equals to the pointwise Jacobian.

We study the validity of the condition (N) of Lusin for homeomorphisms under minimal assumptions on the integrability of Df. It turns out that the role of grand Sobolev spaces W1,n) and Wb 1,n) is crucial. A discussion of bi-Sobolev maps in the plane and their connections with degenerate elliptic PDEs is provided.

Let › ‰ R 2 be a domain. Suppose that f 2 W 1;1 loc (›;R 2 ) is a homeomorphism. Then the components x(w), y(w) of the inverse f i1 = (x;y): › 0 ! › have total variations given by jryj(› 0 ) = › fl fl @f fl fl dz; jrxj(› 0 ) = › fl fl @f @y fl fl dz

Let $$N\ge 3$$ . We construct a homeomorphism $$f$$ in the Sobolev space $$W^{1,1}((0,1)^N,(0,1)^N)$$ such that $$f^{-1}\in W^{1,1}((0,1)^N,(0,1)^N)$$ , $$J_f=0$$ a.e. and $$J_{f^{-1}}=0$$ a.e. It follows that $$f$$ maps a set of full measure to a null set and a remaining null set to a set of full measure.We also show that such a pathological homeomorphism cannot exist in dimension $$N=2$$ or with higher regularity $$f\in W^{1,N-1}$$ .

A chain rule for an a.e. approximately differentiable homeomorphism $${{f: \Omega \subset \mathbb{R}^{n} \xrightarrow{\rm onto}} \Omega^{\prime} \subset \mathbb{R}^{n}}$$ is proved. Namely, there exists a Borel set $${{B \subset \mathcal{R}_{f}}}$$ , where $$\mathcal{R}_{f} = \{x \in \Omega : f\, {\rm is\,approximately\,differentiable\,at}\,x\,{\rm and}\,J_{f}(x) \neq 0\},$$ such that $${{\mid \mathcal{R}_{f} \setminus B \mid = 0}}$$ , $${{f(B) \subset \mathcal{R}_{f-1}}}$$ and J f-1(f(x))J f (x) = 1 for all $${{x \in B}}$$ . In general, it is not true that $${{f(\mathcal{R}_{f}) \subset \mathcal{R}_{f-1}}}$$ . Indeed, there exists a homeomorphism $${f_{0}: \Omega \subset \mathbb{R}^{n} \xrightarrow {\rm onto} \Omega^{\prime} \subset \mathbb{R}^{n}}$$ such that f 0 is approximately differentiable at x 0 with $${{J_{f0}(x_{0}) \neq 0}}$$ and $${{{f_{0}^{-1}}}}$$ is not approximately differentiable at f 0(x 0). We give various conditions to guarantee that a homeomorphism preserves density points.

Function spaces that are slightly larger than the Lebesgue Lp(Ω) spaces (even larger than the Marcinkiewicz Lp,∞(Ω) spaces) have been introduced by Iwaniec and Sbordone [Arch. Ration. Mech. Anal. 119 (1992), 129–143] in connection with integrability properties of the Jacobian. These are the grand Lebesgue spaces Lp)(Ω). In this survey we collect a number of results which prove that these spaces are useful in various classical settings of geometric function theory and partial differential equations (PDEs).

We provide regularity properties for the inverse map f^-1 under suitable assumptions on q-distorsion function of f, in bounded domains of R^2.

We give sharp conditions under which the composition of two homeomorphisms of finite distortion is of finite distortion and has integrable distortion. As an application, we obtain a generalization of the classical uniqueness theorem of homeomorphic solution to the measurable Riemann mapping problem.

Let $p > n-1$ and $\alpha\in\mathsf{R}$ and suppose that $f:\Omega\stackrel{\rm onto\,\,}\longrightarrow\Omega^\prime$ is a homeomorphism in the Zygmund-Sobolev space ${\it WL}^{p}\log^{\alpha} L_{\mathop{\rm loc}\nolimits} (\Omega{,}\mathsf{R}^n)$. Define $r{=}\frac p {p-n+1}$. Assume that $u{\in}{\it WL}^r\log^{-\alpha(r-1)} L_{\mathop{\rm loc}\nolimits}(\Omega)$. Then $u\circ\smash{f^{-1}}\in {\rm BV}_{\mathop{\rm loc}\nolimits}(\Omega^\prime)$. We obtain a similar result whenever $f$ is a homeomorphism in the Lorentz-Sobolev space ${\it WL}^{p,q}_{\mathop{\rm loc}\nolimits} (\Omega,\mathsf{R}^n)$ with $p,q>n-1$ and $u\in {\it WL}^{r,s}_{\mathop{\rm loc}\nolimits}(\Omega)$ with $r$ as before and $s=\frac q {q-n+1}$. Moreover, if we further assume that $f$ has finite inner distortion we obtain in both cases $u\circ \smash{f^{-1}}\in W^{1,1}_{\mathop{\rm loc}\nolimits}(\Omega^\prime)$.