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Register a new userThe square root problem for second order, divergence form operators with mixed boundary conditions on $L^p$
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We show that, under general conditions, the operator $bigl (- abla cdot mu abla +1bigr)^{1/2}$ with mixed boundary conditions provides a topological isomorphism between $W^{1,p}_D(Omega)$ and $L^p(Omega)$, for $p in {]1,2[}$ if one presupposes that this isomorphism holds true for $p=2$. The domain $Omega$ is assumed to be bounded, the Dirichlet part $D$ of the boundary has to satisfy the well-known Ahlfors-David condition, whilst for the points from $overline {partial Omega setminus D}$ the existence of bi-Lipschitzian boundary charts is required.
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We prove a number of textit{a priori} estimates for weak solutions of elliptic equations or systems with vertically independent coefficients in the upper-half space. These estimates are designed towards applications to boundary value problems of Dirichlet and Neumann type in various topologies. We work in classes of solutions which include the energy solutions. For those solutions, we use a description using the first order systems satisfied by their conormal gradients and the theory of Hardy spaces associated with such systems but the method also allows us to design solutions which are not necessarily energy solutions. We obtain precise comparisons between square functions, non-tangential maximal functions and norms of boundary trace. The main thesis is that the range of exponents for such results is related to when those Hardy spaces (which could be abstract spaces) are identified to concrete spaces of tempered distributions. We consider some adapted non-tangential sharp functions and prove comparisons with square functions. We obtain boundedness results for layer potentials, boundary behavior, in particular strong limits, which is new, and jump relations. One application is an extrapolation for solvability a la {v{S}}ne{ui}berg. Another one is stability of solvability in perturbing the coefficients in $L^infty$ without further assumptions. We stress that our results do not require De Giorgi-Nash assumptions, and we improve the available ones when we do so.
Given any elliptic system with $t$-independent coefficients in the upper-half space, we obtain representation and trace for the conormal gradient of solutions in the natural classes for the boundary value problems of Dirichlet and Neumann types with area integral control or non-tangential maximal control. The trace spaces are obtained in a natural range of boundary spaces which is parametrized by properties of some Hardy spaces. This implies a complete picture of uniqueness vs solvability and well-posedness.
We answer the question of when an invariant pseudodifferential operator is Fredholm on a fixed, given isotypical component. More precisely, let $Gamma$ be a compact group acting on a smooth, compact, manifold $M$ without boundary and let $P in psi^m(M; E_0, E_1)$ be a $Gamma$-invariant, classical, pseudodifferential operator acting between sections of two $Gamma$-equivariant vector bundles $E_0$ and $E_1$. Let $alpha$ be an irreducible representation of the group $Gamma$. Then $P$ induces by restriction a map $pi_alpha(P) : H^s(M; E_0)_alpha to H^{s-m}(M; E_1)_alpha$ between the $alpha$-isotypical components of the corresponding Sobolev spaces of sections. We study in this paper conditions on the map $pi_alpha(P)$ to be Fredholm. It turns out that the discrete and non-discrete cases are quite different. Additionally, the discrete abelian case, which provides some of the most interesting applications, presents some special features and is much easier than the general case. We thus concentrate in this paper on the case when $Gamma$ is finite abelian. We prove then that the restriction $pi_alpha(P)$ is Fredholm if, and only if, $P$ is $alpha$-elliptic, a condition defined in terms of the principal symbol of $P$. If $P$ is elliptic, then $P$ is also $alpha$-elliptic, but the converse is not true in general. However, if $Gamma$ acts freely on a dense open subset of $M$, then $P$ is $alpha$-elliptic for the given fixed $alpha$ if, and only if, it is elliptic. The proofs are based on the study of the structure of the algebra $psi^{m}(M; E)^Gamma$ of classical, $Gamma$-invariant pseudodifferential operators acting on sections of the vector bundle $E to M$ and of the structure of its restrictions to the isotypical components of $Gamma$. These structures are described in terms of the isotropy groups of the action of the group $Gamma$ on $E to M$.
Let $p(cdot): mathbb R^nto(0,1]$ be a variable exponent function satisfying the globally log-Holder continuous condition and $L$ a one to one operator of type $omega$ in $L^2({mathbb R}^n)$, with $omegain[0,,pi/2)$, which has a bounded holomorphic functional calculus and satisfies the Davies-Gaffney estimates. In this article, the authors introduce the variable weak Hardy space $W!H_L^{p(cdot)}(mathbb R^n)$ associated with $L$ via the corresponding square function. Its molecular characterization is then established by means of the atomic decomposition of the variable weak tent space $W!T^{p(cdot)}(mathbb R^n)$ which is also obtained in this article. In particular, when $L$ is non-negative and self-adjoint, the authors obtain the atomic characterization of $W!H_L^{p(cdot)}(mathbb R^n)$. As an application of the molecular characterization, when $L$ is the second-order divergence form elliptic operator with complex bounded measurable coefficient, the authors prove that the associated Riesz transform $ abla L^{-1/2}$ is bounded from $W!H_L^{p(cdot)}(mathbb R^n)$ to the variable weak Hardy space $W!H^{p(cdot)}(mathbb R^n)$. Moreover, when $L$ is non-negative and self-adjoint with the kernels of ${e^{-tL}}_{t>0}$ satisfying the Gauss upper bound estimates, the atomic characterization of $W!H_L^{p(cdot)}(mathbb R^n)$ is further used to characterize the space via non-tangential maximal functions.
The aim of the article is to prove $L^{p}-L^{q}$ off-diagonal estimates and $L^{p}-L^{q}$ boundedness for operators in the functional calculus of certain perturbed first order differential operators of Dirac type for with $ple q$ in a certain range of exponents. We describe the $L^{p}-L^{q}$ off-diagonal estimates and the $L^{p}-L^{q}$ boundedness in terms of the decay properties of the related holomorphic functions and give a necessary condition for $L^{p}-L^{q}$ boundedness. Applications to Hardy-Littlewood-Sobolev estimates for fractional operators will be given.
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