Singular Integral Theory Research Articles

Counterexamples and weak (1,1) estimates of wave operators for fourth-order Schrödinger operators in dimension three

This paper is dedicated to investigating the L^{p} -bounds of wave operators W_{\pm}(H,\Delta^{2}) associated with fourth-order Schrödinger operators H=\Delta^{2}+V on \mathbb{R}^{3} with real potentials satisfying |V(x)|\lesssim \langle x\rangle^{-\mu} for some \mu>0 . A recent work by Goldberg and Green (2021) has demonstrated that wave operators W_{\pm}(H,\Delta^{2}) are bounded on L^{p}(\mathbb{R}^{3}) for all 1<p<\infty under the condition that \mu>9 and zero is a regular point of H . In the paper, we aim to further establish endpoint estimates for W_{\pm}(H,\Delta^{2}) in two significant ways. First, we provide counterexamples to illustrate the unboundedness of W_{\pm}(H,\Delta^{2}) on the endpoint spaces L^{1}(\mathbb{R}^{3}) and L^{\infty}(\mathbb{R}^{3}) for non-zero compactly supported potentials V . Second, we establish weak (1,1) estimates for the wave operators W_{\pm}(H,\Delta^{2}) and their dual operators W_{\pm}(H,\Delta^{2})^{*} in the case where zero is a regular point and \mu>11 . These estimates depend critically on the singular integral theory of Calderón–Zygmund on a homogeneous space (X,d\omega) with a doubling measure d\omega .

A note on commutators of singular integrals with BMO and VMO functions in the Dunkl setting

AbstractOn equipped with a root system R, multiplicity function , and the associated measure , we consider a (nonradial) kernel , which has properties similar to those from the classical theory of singular integrals and the Dunkl convolution operator associated with K. Assuming that b belongs to the BMO space on the space of homogeneous type , we prove that the commutator is a bounded operator on for all . Moreover, is compact on , provided . The paper extends results of Han, Lee, Li, and Wick.

Real Analysis, Harmonic Analysis and Applications

The workshop focused on important developments within the last few years in real and harmonic analysis including nonlinear Carleson theorems and singular integral theory, Fourier restriction theory and spherical maximal functions as well as concurrent progress in the application of these for example to partial differential equations.

Operadores integrales singulares, un breve panorama histórico de su evolución

In this work we give an analytical-historical view of the classical theory of singular integrals introduced by A.P. Calderón-A. Zygmund. Emphasis is given to their applications to PDEs and to some projections of the theory.

Product Space Singular Integrals with Mild Kernel Regularity

We develop product space theory of singular integrals with mild kernel regularity. We study these kernel regularity questions specifically in situations that are very tied to the T1 type arguments and the corresponding structural theory. In addition, our results are multilinear.

Singular integrals in quantum Euclidean spaces

We shall establish the core of singular integral theory and pseudodifferential calculus over the archetypal algebras of noncommutative geometry: quantum forms of Euclidean spaces and tori. Our results go beyond Connes’ pseudodifferential calculus for rotation algebras, thanks to a new form of Calderón-Zygmund theory over these spaces which crucially incorporates nonconvolution kernels. We deduceLpL_p-boundedness and Sobolevpp-estimates for regular, exotic and forbidden symbols in the expected ranks. In theL2L_2level both Calderón-Vaillancourt and Bourdaud theorems for exotic and forbidden symbols are also generalized to the quantum setting. As a basic application of our methods, we proveLpL_p-regularity of solutions for elliptic PDEs.

Uniform approximation of functions by solutions of second order homogeneous strongly elliptic equations on compact sets in

We obtain a criterion for the uniform approximability of functions by solutions of second-order homogeneous strongly elliptic equations with constant complex coefficients on compact sets in (the particular case of harmonic approximations is not distinguished). The criterion is stated in terms of the unique (scalar) Harvey–Polking capacity related to the leading coefficient of a Laurent-type expansion (this capacity is trivial in the well-studied case of non-strongly elliptic equations). The proof uses an improvement of Vitushkin’s scheme, special geometric constructions, and methods of the theory of singular integrals. In view of the inhomogeneity of the fundamental solutions of strongly elliptic operators on , the problem considered is technically more difficult than the analogous problem for , .

A criterion for uniform approximability of individual functions by solutions of second-order homogeneous elliptic equations with constant complex coefficients

A natural counterpart of Vitushkin’s criterion is obtained in the problem of uniform approximation of functions by solutions of second-order homogeneous elliptic equations with constant complex coefficient on compact subsets of , . It is stated in terms of a single (scalar) capacity connected with the leading coefficient of the Laurent series. The scheme of approximation uses methods in the theory of singular integrals and, in particular, constructions of certain special Lipschitz surfaces and Carleson measures. Bibliography: 23 titles.

Multilinear operator-valued Calderón-Zygmund theory

We develop a general theory of multilinear singular integrals with operator-valued kernels, acting on tuples of UMD Banach spaces. This, in particular, involves investigating multilinear variants of the R-boundedness condition naturally arising in operator-valued theory. We proceed by establishing a suitable representation of multilinear, operator-valued singular integrals in terms of operator-valued dyadic shifts and paraproducts, and studying the boundedness of these model operators via dyadic-probabilistic Banach space-valued analysis. In the bilinear case, we obtain a T(1)-type theorem without any additional assumptions on the Banach spaces other than the necessary UMD. Higher degrees of multilinearity are tackled via a new formulation of the Rademacher maximal function (RMF) condition. In addition to the natural UMD lattice cases, our RMF condition covers suitable tuples of non-commutative Lp-spaces. We employ our operator-valued theory to obtain new multilinear, multi-parameter, operator-valued theorems in the natural setting of UMD spaces with property α.

Dyadic-probabilistic methods in bilinear analysis

We demonstrate and develop dyadic–probabilistic methods in connection with non-homogeneous bilinear operators, namely singular integrals and square functions. We develop the full non-homogeneous theory of bilinear singular integrals using a modern point of view. The main result is a new global T b Tb theorem for Calderón–Zygmund operators in this setting. Our main tools include maximal truncations, adapted Cotlar type inequalities and suppression and big piece methods. While proving our bilinear results we also advance and refine the linear theory of Calderón–Zygmund operators by improving techniques and results. For example, we simplify and make more efficient some non-homogeneous summing arguments appearing in T 1 T1 type proofs. As a byproduct, we can manage with ease quite general modulus of continuity in the kernel estimates. Our testing conditions are also quite general by virtue of the big piece method of proof.

Bilinear Calderón–Zygmund theory on product spaces

We develop a wide general theory of bilinear bi-parameter singular integrals T. This includes general Calderón–Zygmund type principles in the bilinear bi-parameter setting: easier bounds, like estimates in the Banach range, imply boundedness in the full bilinear range Lp×Lq→Lr, 1/p+1/q=1/r, 1<p,q≤∞, 1/2<r<∞, weighted estimates, mixed-norm estimates, and so on. We build this Calderón–Zygmund theory using the very useful perspective of dyadic representation theorems that hold under testing conditions. New weighted estimates are developed and used effectively throughout. We also develop commutator decompositions and use them to show estimates in the full range for commutators and iterated commutators, like [b1,T]1 and [b2,[b1,T]1]2, where b1 and b2 are little BMO functions. Our commutator method contains several novelties – one is that the new method can be used to simplify and improve linear commutator proofs, even in the two-weight Bloom setting. We also quickly show commutator lower bounds by using and developing the recent median method.

Singular integral operators on tent spaces over spaces of homogeneous type: Fefferman–Stein box maximal functions and pointwise Carleson type estimates

In this article we generalize the singular integral operator theory on weighted tent spaces to spaces of homogeneous type. This generalization of operator theory is in the spirit of C. Fefferman and Stein since we use some auxiliary functionals on tent spaces which play roles similar to the Fefferman–Stein sharp and box maximal functions in the Lebesgue space setting. Our contribution in this operator theory is twofold: for singular integral operators (including maximal regularity operators) on tent spaces pointwise Carleson type estimates are proved and this recovers known results; on the underlying space no extra geometrical conditions are needed and this could be useful for future applications to parabolic problems in rough settings.

Inverse gravimetry: background material and multiscale mollifier approaches

This paper represents an extended version of the publications Freeden (in: Freeden, Nashed, Sonar (eds) Handbook of Geomathematics, 2nd edn, vol 1, Springer, New York, pp 3–78, 2015) and Freeden and Nashed (in: Freeden, Nashed (eds) Handbook of Mathematical Geodesy, Geosystems Mathematics, Birkhauser, Basel, pp 641–685, 2018c). It deals with the ill-posed problem of transferring input gravitational potential information in the form of Newtonian volume integral values to geological output characteristics of the density contrast function. Some essential properties of the Newton volume integral are recapitulated. Different methodologies of the resolution of the inverse gravimetry problem and their numerical implementations are examined including their dependence on the data source. Three types of input information may be distinguished, namely internal (borehole), terrestrial (surface), and/or external (spaceborne) gravitational data sets. Singular integral theory based inversion of the Newtonian integral equation such as a Haar-type solution is handled in a multiscale framework to decorrelate specific geological signal signatures with respect to inherently given features. Reproducing kernel Hilbert space regularization techniques are studied (together with their transition to certain mollified variants) to provide geological contrast density distributions by “downward continuation” from terrestrial and/or spaceborne data. Numerically, reproducing kernel Hilbert space spline solutions are formulated in terms of Gaussian approximating sums for use of gravimeter data systems.

Equivalence of sparse and Carleson coefficients for general sets

We remark that sparse and Carleson coefficients are equivalent for every countable collection of Borel sets and hence, in particular, for dyadic rectangles, the case relevant to the theory of bi-parameter singular integrals. The key observation is that a dual refomulation by I. E. Verbitsky for Carleson coefficients over dyadic cubes holds also for Carleson coefficients over general sets. We give a simple proof for this reformulation.

Integral equations of the third kind for the case of piecewise monotone coefficients

We examine the third kind integral equations in Hölder class. The coefficients of the equations are piecewise strictly monotone functions having simple zeros. By singular integral equations theory, for solvability of considered equations, we give the necessary and sufficient conditions. Finding a solution is reduced to solving a regular integral equation of second kind.

Hardy-Type Spaces Arising from a Vector-Valued Cauchy Kernel

An integral formula of Cauchy type was recently developed that reproduces any continuous \(f:\overline{{\mathbb {B}}} \rightarrow {\mathbb {C}}^n\) that is holomorphic in the open unit ball \({\mathbb {B}}\) of \({\mathbb {C}}^n\) using a fixed vector-valued kernel and the scalar expression \(\langle f(u),u \rangle \), where \(u\in \partial {\mathbb {B}}\) and \(\langle \cdot ,\cdot \rangle \) is the Hermitian inner product in \({\mathbb {C}}^n\), which is key to defining the numerical range of f. We consider Hardy-type spaces associated with this vector-valued kernel. In particular, we introduce spaces of vector-valued holomorphic mappings properly containing the vector-valued Hardy spaces that are reproduced through the process described above and isomorphic spaces of scalar-valued non-holomorphic functions that satisfy many of the familiar properties of Hardy space functions. In the spirit of providing a straightforward introduction to these spaces, proof techniques have been kept as elementary as possible. In particular, the theory of maximal functions and singular integrals is avoided.

An Element-Driven Boundary Integral Treatment for Nonlinearity: A Coupled Nonlinear Two-Dimensional Burger’s Equation Test Case

To further improve on the competitiveness of the boundary element method (BEM), a hybrid version of it is used for a numerical solution of two dimensional nonlinear coupled viscous Burger’s equation. Adopting this approach to a discretized 2D spatial domain, the resulting integral equations arising from the singular integral theory are applied locally to each of the elements. The resulting nonlinear discrete equations are finally solved by the Picard iteration algorithm. The simulation results obtained, not only concur with analytical solutions, but also display high accuracy and are in agreement with those available in literature.

Removable singularities for the Von Karman equations

A theorem for removable singularities of the fourth order system of equations in the plane called the Von Karman equations is established. This result is best possible in several different ways. The key idea is to apply the Calderon and Zygmund singular integral theory in conjunction with several lemmas to a nontrivial bootstrap argument.

On uniqueness of weak solutions for the thin-film equation

In any number of space variables, we study the Cauchy problem related to the fourth-order degenerate diffusion equation ∂sh+∇⋅(h∇Δh)=0. This equation, derived from a lubrication approximation, also models the surface tension dominated flow of a thin viscous film in the Hele-Shaw cell. Our focus is on uniqueness of weak solutions in the “complete wetting” regime, when a zero contact angle between liquid and solid is imposed. In this case, we transform the problem by zooming into the free boundary and look at small Lipschitz perturbations of a quadratic stationary solution. In the perturbational setting, the main difficulty is to construct scale invariant function spaces in such a way that they are compatible with the structure of the nonlinear equation. Here we rely on the theory of singular integrals in spaces of homogeneous type to obtain linear estimates in these functions spaces which provide “optimal” conditions on the initial data under which a unique solution exists. In fact, this solution can be used to define a class of functions in which the original initial value problem has a unique (weak) solution. Moreover, we show that the interface between empty and wetted regions is an analytic hypersurface in time and space.

𝐿^{𝑝} theory for outer measures and two themes of Lennart Carleson united

We develop a theory of L p L^p spaces based on outer measures generated through coverings by distinguished sets. The theory includes as a special case the classical L p L^p theory on Euclidean spaces as well as some previously considered generalizations. The theory is a framework to describe aspects of singular integral theory, such as Carleson embedding theorems, paraproduct estimates, and T ( 1 ) T(1) theorems. It is particularly useful for generalizations of singular integral theory in time-frequency analysis, the latter originating in Carleson’s investigation of convergence of Fourier series. We formulate and prove a generalized Carleson embedding theorem and give a relatively short reduction of the most basic L p L^p estimates for the bilinear Hilbert transform to this new Carleson embedding theorem.