Estimates For Singular Integrals Research Articles

Global strong solutions to nonlocal Benjamin-Bona-Mahony equations with exponential nonlinearities

In the current work, we consider the Cauchy problem for a class of adjusted Benjamin-Bona-Mahony (BBM) equations. These equations are modified by considering the time-fractional Caputo derivative of order α∈(0,1) (instead of the classical one) and an additional nonlinearity of exponential type. The first main result includes the unique global existence of strong solutions. The approach for this goal can be summarized as follows. First of all, we use the standard contraction arguments to prove the local existence and uniqueness of a mild solution. Next, apply a weak version of Grönwall's inequality to improve the temporal regularity of the solutions. Using this regularity, we deduce energy estimates for solutions which helps us to obtain the global boundedness. The second aim of the study is about the behavior of solutions according to the fractional order α. Precisely, we show that our solutions converge to those of the classical model (with integer order derivative) as α approaches 1−. The desired result is derived by some singular integral estimates which is the combination of some essential basic inequalities.

Sharp Trace and Korn Inequalities for Differential Operators

AbstractWe establish sharp trace and Korn-type inequalities that involve vectorial differential operators, the focus being on situations where global singular integral estimates are not available. Starting from a novel approach to sharp Besov boundary traces by Riesz potentials and oscillations that equally applies to $$p=1$$ p = 1 , a case difficult to be handled by harmonic analysis techniques, we then classify boundary trace- and Korn-type inequalities. For $$p=1$$ p = 1 and so despite the failure of the Calderón-Zygmund theory, we prove that sharp trace estimates can be systematically reduced to full k-th order gradient estimates. Moreover, for $$1<p<\infty $$ 1 < p < ∞ , where sharp trace estimates yield Korn-type inequalities on smooth domains, we show for the basically optimal class of John domains that Korn-type inequalities persist – even though the reduction to global Calderón-Zygmund estimates by extension operators might not be possible.

Causal sparse domination of Beurling maximal regularity operators

We prove boundedness of Calderön–Zygmund operators acting in Banach function spaces on domains, defined by the L1 Carleson functional and Lq (1 < q < ∞) Whitney averages. For such bounds to hold, we assume that the operator maps towards the boundary of the domain. We obtain the Carleson estimates by proving a pointwise domination of the operator, by sparse operators with a causal structure. The work is motivated by maximal regularity estimates for elliptic PDEs and is related to one-sided weighted estimates for singular integrals.

Weighted BMO estimates for singular integrals and endpoint extrapolation in Banach function spaces

In this paper we prove sharp weighted BMO estimates for singular integrals, and we show how such estimates can be extrapolated to Banach function spaces.

Global estimates for the fundamental solution of homogeneous Hörmander operators

Let $${\mathcal {L}}=\sum _{j=1}^{m}X_{j}^{2}$$ be a Hörmander sum of squares of vector fields in $${\mathbb {R}}^{n}$$ , where any $$X_{j}$$ is homogeneous of degree 1 with respect to a family of non-isotropic dilations in $${\mathbb {R}}^{n}$$ . Then, $${\mathcal {L}}$$ is known to admit a global fundamental solution $$\Gamma (x;y)$$ that can be represented as the integral of a fundamental solution of a sublaplacian operator on a lifting space $${\mathbb {R}}^{n}\times {\mathbb {R}}^{p}$$ , equipped with a Carnot group structure. The aim of this paper is to prove global pointwise (upper and lower) estimates of $$\Gamma$$ , in terms of the Carnot–Carathéodory distance induced by $$X=\{X_{1},\ldots ,X_{m}\}$$ on $${\mathbb {R}}^{n}$$ , as well as global pointwise (upper) estimates for the X-derivatives of any order of $$\Gamma$$ , together with suitable integral representations of these derivatives. The least dimensional case $$n=2$$ presents several peculiarities which are also investigated. Applications to the potential theory for $${\mathcal {L}}$$ and to singular-integral estimates for the kernel $$X_{i}X_{j}\Gamma$$ are also provided. Finally, most of the results about $$\Gamma$$ are extended to the case of Hörmander operators with drift $$\sum _{j=1}^{m}X_{j}^{2}+X_{0}$$ , where $$X_{0}$$ is 2-homogeneous and $$X_{1},\ldots ,X_{m}$$ are 1-homogeneous.

Genuinely multilinear weighted estimates for singular integrals in product spaces

We prove genuinely multilinear weighted estimates for singular integrals in product spaces. The estimates complete the qualitative weighted theory in this setting. Such estimates were previously known only in the one-parameter situation. Extrapolation gives powerful applications – for example, a free access to mixed-norm estimates in the full range of exponents.

Asymptotic Estimates for Singular Integrals of Fractions Whose Denominators Contain Products of Block Quadratic Forms

Asymptotic Estimates for Singular Integrals of Fractions Whose Denominators Contain Products of Block Quadratic Forms

Critical weak-$L^p$ differentiability of singular integrals

We establish that for every function u \in L^1_{\rm loc}(\Omega) whose distributional Laplacian \Delta u is a signed Borel measure in an open set \Omega in \mathbb{R}^{N} , the distributional gradient \nabla u is differentiable almost everywhere in \Omega with respect to the weak- L^{N/(N-1)} Marcinkiewicz norm. We show in addition that the absolutely continuous part of \Delta u with respect to the Lebesgue measure equals zero almost everywhere on the level sets \{u= \alpha\} and \{\nabla u=e\} , for every \alpha \in \mathbb{R} and e \in \mathbb{R}^N . Our proofs rely on an adaptation of Calderón and Zygmund's singular-integral estimates inspired by subsequent work by Hajlasz.

L1-Poincaré inequalities for differential forms on Euclidean spaces and Heisenberg groups

In this paper, we prove interior Poincaré and Sobolev inequalities in Euclidean spaces and in Heisenberg groups, in the limiting case where the exterior (resp. Rumin) differential of a differential form is measured in L1 norm. Unlike for Lp, p>1, the estimates are doomed to fail in top degree. The singular integral estimates are replaced with inequalities which go back to Bourgain-Brezis in Euclidean spaces, and to Chanillo-Van Schaftingen in Heisenberg groups.

L1-Poincaré and Sobolev inequalities for differential forms in Euclidean spaces

In this paper, we prove Poincare and Sobolev inequalities for differential forms in L1(ℝn). The singular integral estimates that it is possible to use for Lp, p > 1, are replaced here with inequalities which go back to Bourgain and Brezis (2007).

Interior regularity for degenerate equations with drift on homogeneous groups

Let \(X_0,X_1,\ldots ,X_m\) be left invariant real smooth vector fields on the homogeneous group \(\mathbb {G}=(\mathbb {R}^N,\circ ,\delta _\lambda )\) satisfying the Hormander condition. Assume that \(X_1,\ldots ,X_m\)\((m\le N-1)\) are homogeneous of degree one and \(X_0\) is homogeneous of degree two with respect to the family of dilations \((\delta _\lambda )_{\lambda >0}\). Consider the second order degenerate equation with drift on \(\mathbb {G}\) $$\begin{aligned} \sum _{i,j=1}^{m}X_i(a_{ij}(x)X_ju)+X_0u=\sum _{j=1}^{m}X_jF_j(x), \end{aligned}$$ where the matrix \(\{a_{ij} (x)\}\) of coefficients is symmetric positive definite on \(\mathbb {R}^m\) belonging to the VMO space defined on \(\mathbb {G}\). We prove a Sobolev type inequality for weak solutions to the equation and then the higher \(L^p\) estimates for the gradients of weak solutions. Interior Morrey estimates and Holder continuity for the weak solutions are derived by using the representation formula of weak solutions and estimates of singular integrals.

$L^1$-Dini conditions and limiting behavior of weak type estimates for singular integrals

Let T_\Omega be the singular integral operator with a homogeneous kernel \Omega . In 2006, Janakiraman showed that if \Omega has mean value zero on \mathbb S^{n-1} and satisfies the condition (\ast)\quad \sup_{|\xi|=1}\int_{\S^{n-1}}|\Omega(\theta)-\Omega(\theta+\delta\xi)|\,d\sigma(\theta)\leq Cn\,\delta\int_{\mathbb{S}^{n-1}}|\Omega(\theta)|\,d\sigma(\theta), where 0&lt;\delta&lt;{1}/{n} , then the following limiting behavior: (\ast\ast)\quad \lim\limits_{\lambda\to 0_+}\lambda \, m(\{x\in\mathbb R^n:|T_\Omega f(x)|&gt;\lambda\})= \frac{1}{n}\,\|\Omega\|_{1}\|f\|_{1} holds for f\in L^1(\mathbb R^n) and f\geq 0 . In the present paper, we prove that if we replace the condition (\ast) by a more general condition, the L^1 -Dini condition, then the limiting behavior (\ast\ast) still holds for the singular integral T_\Omega . In particular, we give an example which satisfies the L^1 -Dini condition, but does not satisfy (\ast) . Hence, we improve essentially Janakiraman's above result. To prove our conclusion, we show that the L^1 -Dini conditions defined respectively via rotation and translation in \mathbb R^n are equivalent (see Theorem 2.5 below), which may have its own interest in the theory of the singular integrals. Moreover, similar limiting behavior for the fractional integral operator T_{\Omega,\alpha} with a homogeneous kernel is also established in this paper.

Weak Estimates of Singular Integrals with Variable Kernel and Fractional Differentiation on Morrey-Herz Spaces

Let T be the singular integral operator with variable kernel defined by Tf(x)=p.v.∫Rn(Ω(x,x-y)/x-yn)f(y)dy and let Dγ (0≤γ≤1) be the fractional differentiation operator. Let T⁎and T♯ be the adjoint of T and the pseudoadjoint of T, respectively. In this paper, the authors prove that TDγ-DγT and (T⁎-T♯)Dγ are bounded, respectively, from Morrey-Herz spaces MK˙p,1α,λ(Rn) to the weak Morrey-Herz spaces WMK˙p,1α,λ(Rn) by using the spherical harmonic decomposition. Furthermore, several norm inequalities for the product T1T2 and the pseudoproduct T1∘T2 are also given.

Singular integrals, maximal functions and Fourier restriction to spheres: The disk multiplier revisited

Several estimates for singular integrals, maximal functions and spherical summation operator are given in the spaces LradpLang2(Rn), n≥2.

Limiting Bourgain–Brezis estimates for systems of linear differential equations: Theme and variations

J. Bourgain and H. Brezis have obtained in 2002 some new and surprising estimates for systems of linear differential equations, dealing with the endpoint case L¹ of singular integral estimates and the critical Sobolev space W1,n(Rn). This paper presents an overview of the results, further developments over the last ten years and challenging open problems.

Strong and Weak Type Estimates for Singular Integrals with Respect to Measures Separated by AD-regular Boundaries

A Radon measure on R has n-growth if there exists some constant cμ such that μ(B(x, r)) ≤ cμr n for all x ∈ R, r > 0. If there exists some constant cμ such that c μ r n ≤ μ(B(x, r)) ≤ cμ r n for all x ∈ sptμ, 0 0 and η, with 0 < η ≤ 1, such that the following inequalities hold for all x, y ∈ R, x 6= y:

An extrapolation theorem with applications to weighted estimates for singular integrals

We prove an extrapolation theorem saying that the weighted weak type (1,1) inequality for A1 weights implies the strong Lp(w) bound in terms of the Lp(w) operator norm of the maximal operator M. The weak Muckenhoupt–Wheeden conjecture along with this result allows us to conjecture that the following estimate holds for a Calderón–Zygmund operator T for any p>1:‖T‖Lp(w)⩽c‖M‖Lp(w)p. The latter conjecture would yield the sharp estimates for ‖T‖Lp(w) in terms of the Aq characteristic of w for any 1<q<p. In this paper we get a weaker inequality‖T‖Lp(w)⩽c‖M‖Lp(w)plog(1+‖M‖Lp(w)) with the corresponding estimates for ‖w‖Aq when 1<q<p.

Some strong [formula omitted]-type inequalities for the homotopy operator

We prove the boundedness of the homotopy operator in the weighted L p spaces. As extensions of ( p , p ) -type norm inequalities for the homotopy operator, we also establish some strong ( p , q ) -type norm inequalities with A r weights and power-type weights for the homotopy operator acting on differential forms. Finally, we obtain some estimates for singular integrals.

Global Estimates for Singular Integrals of the Composition of the Maximal Operator and the Green's Operator

We establish the Poincaré type inequalities for the composition of the maximal operator and the Green's operator in John domains.

Weighted rearrangement inequalities for local sharp maximal functions

Several weighted rearrangement inequalities for uncentered and centered local sharp functions are proved. These results are applied to obtain new weighted weak-type and strong-type estimates for singular integrals. A self-improving property of sharp function inequalities is established.