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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.

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.

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.

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 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.

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]

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]

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]

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]

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]

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]

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).

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]

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]

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]

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.

It is known that functions in a Sobolev space with critical exponent embed into the space of functions of bounded mean oscillation, and therefore satisfy the John-Nirenberg inequality and a corresponding exponential integrability estimate. While these inequalities are optimal for general functions of bounded mean oscillation, the main result of this paper is an improvement for functions in a class of critical Sobolev spaces. Precisely, we prove the inequality

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.

It was conjectured by M. Glasser and S. Davison and later proved by A.Eremenko that the certain animals should gather close to each other in order todecrease the total heat loss. In this paper we show that it is not always truefor the individual heat loss. This gives a negative answer to a question posedby A. Eremenko.

A method for the kernel-independent construction of H 2 -matrix approximations to non-local operators is proposed. Special attention is paid to the adaptive construction of nested bases. As a side result, new error estimates for adaptive cross approximation (ACA) are presented which have implications on the pivoting strategy of ACA. [ABSTRACT FROM AUTHOR]

In this paper, generalized Bessel potential is studied. A generalization is achieved by considering the singular Laplace–Bessel operator instead of the Laplace operator in the construction of the Bessel potential. First, we give some auxiliary statements as weighted plane wave relation and generalized Bochner formula. Next, we consider the Bessel kernel and prove some of its properties. After that, we present the definition and properties of generalized Bessel potential. Finally, we consider the generalized Bessel potential as a solution to the singular screened Poisson equation. [ABSTRACT FROM AUTHOR]

In this paper we consider classes of initial data that ensure local-in-time Hadamard well-posedness of the associated weak Leray–Hopf solutions of the three-dimensional Navier–Stokes equations. In particular, for any solenodial L 2 initial data u 0 belonging to certain subsets of V M O - 1 (R 3) , we show that weak Leray–Hopf solutions depend continuously with respect to small divergence-free L 2 perturbations of the initial data u 0 (on some finite-time interval). Our main result is inspired by and improves upon previous work of the author (Barker in J Math Fluid Mech 20(1):133–160, 2018) and work of Jean–Yves Chemin (Commun Pure Appl Math 64(12):1587–1598, 2011). Our method builds upon [4, 9]. In particular our method hinges on decomposition results for the initial data inspired by Calderón (Trans Am Math Soc 318(1):179–200, 1990) together with use of persistence of regularity results. The persistence of regularity statement presented may be of independent interest, since it does not rely upon the solution or the initial data being in the perturbative regime. [ABSTRACT FROM AUTHOR]

We consider the existence of positive solutions to weighted quasilinear elliptic differential equations of the type - Δ p , w u = σ u q in Ω , u = 0 on ∂ Ω in the sub-natural growth case 0 < q < p - 1 , where Ω is a bounded domain in R n , Δ p , w is a weighted p-Laplacian, and σ is a nonnegative (locally finite) Radon measure on Ω . We give criteria for the existence problem. For the proof, we investigate various properties of p-superharmonic functions, especially the solvability of Dirichlet problems with infinite measure data. [ABSTRACT FROM AUTHOR]

This paper proposes a fairly general new point of view on the question of asymptotic stability of (topological) solitons. Our approach is based on the use of the distorted Fourier transform at the nonlinear level; it does not rely only on Strichartz or virial estimates and is therefore able to treat low-power nonlinearities (hence also nonlocalised solitons) and capture the global (in space and time) behaviour of solutions.

This paper is devoted to a simple proof of the generalized Leibniz rule in bounded domains. The operators under consideration are the so-called spectral Laplacian and the restricted Laplacian. Equations involving such operators have lately been considered by Constantin and Ignatova in the framework of the SQG equation [P. Constantin and M. Ignatova, Critical SQG in bounded domains, Ann. PDE 2 2016, 2, Article ID 8] in bounded domains, and by two of the authors [Q.-H. Nguyen and J. L. Vázquez, Porous medium equation with nonlocal pressure in a bounded domain, Comm. Partial Differential Equations 43 2018, 10, 1502–1539] in the framework of the porous medium with nonlocal pressure in bounded domains. We will use the estimates in this work in a forthcoming paper on the study of porous medium equations with pressure given by Riesz-type potentials. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to find pointwise estimates on the integral kernel for the □ b Green operator, N, for a particular quadric submanifold of codimension 2 in C 4 . This particular kernel has singularities away from the diagonal. Furthermore, the estimates exhibited by this kernel, both on and off the diagonal, follow no known paradigm. [ABSTRACT FROM AUTHOR]

Let L 2 = (- Δ) 2 + V 2 be the Schrödinger-type operator on R n (n ≥ 5) , where the nonnegative potential V belongs to the reverse Hölder class R H s , n / 2 < s < n. The estimates of L p (R n) and weak type (1, 1) for the Riesz transform ∇ L 2 - 1 4 related to Schrödinger-type operators are established. [ABSTRACT FROM AUTHOR]

In this article, we obatin the l ∞ estimate of the kernel a n , m (t) for m = 0 , 1 , m = n and t ∈ [ 1 , ∞ ] for the propagator e - i t H d of one dimensional difference operator associated with the Hermite functions. We conjecture that this estimate holds true for any positive integer m and in that case, we obtain better decay for ‖ e - i t H d ‖ l 1 → l ∞ and ‖ e - i t H d ‖ l σ 2 → l - σ 2 for large |t| compare to the Euclidean case, see Egorova (J Spectr Theory 5:663–696, 2015). These estimates are useful in the analysis of one-dimensional discrete Schrödinger equation associated with operator H d . [ABSTRACT FROM AUTHOR]

For a normalized root system R in R N and a multiplicity function k ≥ 0 let N = N + ∑ α ∈ R k (α) . We denote by d w (x) = Π α ∈ R | ⟨ x , α ⟩ | k (α) d x the associated measure in R N . Let L = - Δ + V , V ≥ 0 , be the Dunkl–Schrödinger operator on R N . Assume that there exists q > max (1 , N 2) such that V belongs to the reverse Hölder class RH q (d w) . For λ > 0 we provide upper and lower estimates for the number of eigenvalues of L which are less or equal to λ . Our main tool in the Fefferman–Phong type inequality in the rational Dunkl setting. [ABSTRACT FROM AUTHOR]

We present recent results on elliptic boundary value problems where the theory of Hardy spaces associated with operators plays a key role. [ABSTRACT FROM AUTHOR]