Maximal Singular Integrals Research Articles

Multiple weighted estimates for maximal vector-valued commutator of multilinear Calderón-Zygmund singular integrals

This paper concerns with multiple weighted norm inequalities for maximal vector-valued multilinear singular operator and maximal commutators. The Cotlar-type inequality of maximal vector-valued multilinear singular integrals operator is obtained. On the other hand, pointwise estimates for sharp maximal function of two kinds of maximal vector-valued multilinear singular integrals and maximal vector-valued commutators are also established. By the weighted estimates of a class of new variant maximal operator, Cotlar's inequality and the sharp maximal function estimates, multiple weighted strong estimates and weak estimates for maximal vector-valued singular integrals of multilinear operators and those for maximal vector-valued commutator of multilinear singular integrals are obtained.

L p estimates of rough maximal functions along surfaces with applications

In this paper, we study the Lp mapping properties of certain class of maximal oscillatory singular integral operators. We prove a general theorem for a class of maximal functions along surfaces. As a consequence of such theorem, we establish the Lp boundedness of various maximal oscillatory singular integrals provided that their kernels belong to the natural space Llog L(Sn−1). Moreover, we highlight some additional results concerning operators with kernels in certain block spaces. The results in this paper substantially improve previously known results.

Sharp weighted bounds for multiple integral operators

Sharp weighted bounds for strong maximal functions, multiple potentials and singular integrals are derived in terms of Muckenhoupt type characteristics of weights.

L P estimates for the maximal singular integral in terms of the singular integral

This paper continues the study, initiated in [MOV] and [MOPV], of the problem of controlling the maximal singular integral T* f by the singular integral Tf. Here, T is a smooth homogeneous Calderon-Zygmund singular integral operator of convolution type. We consider two forms of control, namely, in the weighted Lp(ω) norm and via pointwise estimates of T* f by M(Tf ) or M2(Tf), where M is the Hardy-Littlewood maximal operator and M2 = M po M its iteration. The novelty with respect to the aforementioned works lies in the fact that here p is different from 2 and the Lp space is weighted.

The [formula omitted] endpoint estimate for maximal singular integral operators

We prove in this paper the following estimate for the maximal operator T⁎ associated to the singular integral operator T:‖T⁎f‖L1,∞(w)≲1ϵ∫Rn|f(x)|ML(log⁡L)ϵ(w)(x)dx,w≥0,0<ϵ≤1. This follows from the sharp Lp estimate‖T⁎f‖Lp(w)≲p′(1δ)1/p′‖f‖Lp(ML(log⁡L)p−1+δ(w)),1<p<∞,w≥0,0<δ≤1. As a consequence we deduce that‖T⁎f‖L1,∞(w)≲[w]A1log⁡(e+[w]A∞)∫Rn|f|wdx, extending the endpoint results obtained in [23] and [15] to maximal singular integrals. Another consequence is a quantitative two weight bump estimate.

𝐿^{𝑝} bounds for the commutators of singular integrals and maximal singular integrals with rough kernels

The commutator of convolution type Calderon-Zygmund singular integral operators with rough kernels p . v . Ω ( x ) | x | n p.v. \frac {\Omega (x)}{|x|^n} are studied. The authors established the L p ( 1 &gt; p &gt; ∞ ) L^p\,(1&gt;p&gt;\infty ) boundedness of the commutators of singular integrals and maximal singular integrals with the kernel condition which is different from the condition Ω ∈ H 1 ( S n − 1 ) . \Omega \in H^1(S^{n-1}).

L^P-ESTIMATES FOR THE ROUGH SINGULAR INTEGRALS ASSOCIATED TO SURFACES

In this paper we obtain the $L^{p}$-boundedness for the maximal functions and the singular integrals associated to surfaces $(y,\phi(\lvert y\rvert))$ with rough kernels, $1 < p < \infty$. The analogue estimate is also established for the corresponding maximal singular integrals.

Parabolic singular integrals with rough kernels and extrapolation

This paper is devoted to studying the parabolic singular integrals with rough kernels both on the unit sphere and in the radial direction, as well as the corresponding maximal singular integrals. By the estimates of Fourier transforms, the Littlewood-Paley theory and the extrapolation arguments, under the rather weak size conditions, which are the best known ones so far, the L p bounds for such operators are established. As applications, the corresponding results for the corresponding operators associated to certain smooth surfaces are also obtained.

Maximal singular integrals on product homogeneous groups

Maximal singular integrals on product homogeneous groups

On joint estimates for maximal functions and singular integrals on weighted spaces

We consider a conjecture attributed to Muckenhoupt and Wheeden which suggests a positive relationship between the continuity of the Hardy-Littlewood maximal operator and the Hilbert transform in the weighted setting. Although continuity of the two operators is equivalent for A p A_p weights with 1 &gt; p &gt; ∞ 1 &gt; p &gt; \infty , through examples we illustrate this is not the case in more general contexts. In particular, we study weights for which the maximal operator is bounded on the corresponding L p L^p spaces while the Hilbert transform is not. We focus on weights which take the value zero on sets of nonzero measure and exploit this lack of strict positivity in our constructions. These types of weights and techniques have been explored previously by the first author and independently with C. Thiele.

Logarithmic L p Bounds for Maximal Directional Singular Integrals in the Plane

Let K be a Calderon–Zygmund convolution kernel on ℝ. We discuss the L p -boundedness of the maximal directional singular integral $$T_{\mathbf{V}} f (x)= \sup_{v \in \mathbf{V}} \bigg| \int_{\mathbb{R}} f(x+t v) K(t) \, \mathrm{d} {t} \bigg| $$ where V is a finite set of N directions. Logarithmic bounds (for 2≤p<∞) are established for a set V of arbitrary structure. Sharp bounds are proved for lacunary and Vargas sets of directions. The latter include the case of uniformly distributed directions and the finite truncations of the Cantor set. We make use of both classical harmonic analysis methods and product-BMO based time-frequency analysis techniques. As a further application of the latter, we derive an L p almost orthogonality principle for Fourier restrictions to cones.

A characterization of two weight norm inequalities for maximal singular integrals with one doubling measure

We characterize two-weight inequalities for certain maximal truncations of the Hilbert transform in terms of testing conditions on simpler functions. For 1<p<2 and two positive Borel measures u, v on R, we assume that u is doubling, and we consider maximal truncations T_# of the Hilbert transform. The norm estimate || T(f u) ||_{L^p(v)} < C || f ||_{L^p(u)} is characterized in terms of an A_p condition on the weights and two testing conditions. The first is the norm condition above, but the function f varies over bounded functions supported on a cube. The second is a dual weak-type condition, for arbitrary functions. This result should be compared to the result of Nazarov, Treil and Volberg, arXiv:math/0702758. Additional results are obtained for 2<p<\infty, and for the weak type inequality.

The boundedness of maximal operators and singular integrals via Fourier transform estimates

In this paper, the author studies the mapping properties for some general maximal operators and singular integrals on certain function spaces via Fourier transform estimates. Also, some concrete maximal operators and singular integrals are studied as applications.

Estimates for the maximal singular integral in terms of the singular integral: the case of even kernels

Let T be a smooth homogeneous Calder on-Zygmund singular integral operator in R n . In this paper we study the problem of controlling the maximal singular integral T ? f by the singular integral Tf. The most basic form of control one may consider is the estimate of the L 2 (R n ) norm of T ? f by a constant times the L 2 (R n ) norm of Tf. We show that if T is an even higher order Riesz transform, then one has the stronger pointwise inequality T ? f(x) C M(Tf)(x), where C is a constant and M is the Hardy-Littlewood maximal operator. We prove that the L 2 estimate of T ? by T is equivalent, for even smooth homogeneous Calder on-Zygmund operators, to the pointwise inequality between T ? and M(T ). Our main result characterizes the L 2 and pointwise inequalities in terms of an algebraic condition expressed in terms of the kernel ( x) jxjn of T , where is an

Sharp weighted estimates for classical operators

We give a general method based on dyadic Calderón–Zygmund theory to prove sharp one- and two-weight norm inequalities for some of the classical operators of harmonic analysis: the Hilbert and Riesz transforms, the Beurling–Ahlfors operator, the maximal singular integrals associated to these operators, the dyadic square function and the vector-valued maximal operator. In the one-weight case we prove the sharp dependence on the A p constant by finding the best value for the exponent α ( p ) such that ‖ T f ‖ L p ( w ) ⩽ C n , T [ w ] A p α ( p ) ‖ f ‖ L p ( w ) . For the Hilbert transform, the Riesz transforms and the Beurling–Ahlfors operator the sharp value of α ( p ) was found by Petermichl and Volberg (2007, 2008, 2002) [47–49]; their proofs used approximations by the dyadic Haar shift operators, Bellman function techniques, and two-weight norm inequalities. Our proofs again depend on dyadic approximation, but avoid Bellman functions and two-weight norm inequalities. We instead use a recent result due to A. Lerner (2010) [34] to estimate the oscillation of dyadic operators. By applying this we get a straightforward proof of the sharp dependence on the A p constant for any operator that can be approximated by Haar shift operators. In particular, we provide a unified approach for the Hilbert and Riesz transforms, the Beurling–Ahlfors operator (and their corresponding maximal singular integrals), dyadic paraproducts and Haar multipliers. Furthermore, we completely solve the open problem of sharp dependence for the dyadic square functions and vector-valued Hardy–Littlewood maximal function. In the two-weight case we use the very same techniques to prove sharp results in the scale of A p bump conditions. For the singular integrals considered above, we show they map L p ( v ) into L p ( u ) , 1 < p < ∞ , if the pair ( u , v ) satisfies sup Q ‖ u 1 / p ‖ A , Q ‖ v − 1 / p ‖ B , Q < ∞ , where A ¯ ∈ B p ′ and B ¯ ∈ B p are Orlicz functions. This condition is sharp. Furthermore, this condition characterizes (in the scale of these A p bump conditions) the corresponding two-weight norm inequality for the Hardy–Littlewood maximal operator M and its dual: i.e., M : L p ( v ) → L p ( u ) and M : L p ′ ( u 1 − p ′ ) → L p ( v 1 − p ′ ) . Muckenhoupt and Wheeden conjectured that these two inequalities for M are sufficient for the Hilbert transform to be bounded from L p ( v ) into L p ( u ) . Thus, in the scale of A p bump conditions, we prove their conjecture. We prove similar, sharp two-weight results for the dyadic square function and the vector-valued maximal operator.

Multiple-weighted norm inequalities for maximal multi-linear singular integrals with non-smooth kernels

We obtain weighted norm inequalities for maximal truncated operators of multi-linear singular integrals with non-smooth kernels in the sense of Duong et al. This class of operators extends the class of multi-linear Calderón-Zygmund operators introduced by Coifman and Meyer and includes the higher-order commutators of Calderón. The weighted norm inequalities obtained in this work are with respect to the new class of multiple weights of Lerner et al. The key ingredient in the proof is the introduction of a new multi-sublinear maximal operator that plays the role of the Hardy-Littlewood maximal function in a version of Cotlar's inequality. As applications of these results, new weighted estimates for the mth order Calderón commutators and their maximal counterparts are deduced.

Equivalent Characterizations for Boundedness of Maximal Singular Integrals on ax+b-Groups

Let (S,d,ρ) be the affine group ℝn⋉ℝ+ endowed with the left-invariant Riemannian metric d and the right Haar measure ρ, which is of exponential growth at infinity. In this paper, for any linear operator T on (S,d,ρ) associated with a kernel K satisfying certain integral size condition and Hörmander’s condition, the authors prove that the following four statements regarding the corresponding maximal singular integral T ∗ are equivalent: T ∗ is bounded from \(L_{c}^{\infty}\) to BMO, T ∗ is bounded on L p for all p∈(1,∞), T ∗ is bounded on L p for some p∈(1,∞) and T ∗ is bounded from L 1 to L 1,∞. As applications of these results, for spectral multipliers of a distinguished Laplacian on (S,d,ρ) satisfying certain Mihlin-Hörmander type condition, the authors obtain that their maximal singular integrals are bounded from \(L_{c}^{\infty}\) to BMO, from L 1 to L 1,∞, and on L p for all p∈(1,∞).

Two–weight inequalities for fractional maximal functions and singular integrals in L p(·) spaces

Various type of two–weight criteria for fractional maximal operators (involving the Hardy–Littlewood maximal functions) and singular integrals in variable exponent Lebesgue spaces defined on the real line are established. Bibliography: 30 titles.

On the boundedness of a class of rough maximal operators on product spaces

In this paper, we study the Lp boundedness of a class of maximal operators T{Ωj}(γ) and a related class of rough singular integrals on product spaces. We obtain appropriate Lp estimates for such maximal operators and singular integrals. These estimates are used in an extrapolation argument and allow us to obtain some new and improved results for certain maximal integral operators and singular integrals on product spaces under certain sharp conditions on the kernel functions. Also, one of our main results in this paper is a corrigendum of a result obtained by Ding-Lin.

Maximal operators and singular integrals with non-isotropic dilation on product domains

National Natural Science Foundation of China [10771054, 10971141]; NSF of Beijing [1092004]