Strong Type Estimates Research Articles

Weak and strong type estimates for square functions associated with operators

Let L be a linear operator on L2(Rn) which generates a semigroup e−tL whose kernels pt(x,y) satisfy the Gaussian upper bound. In this paper, we investigate several kinds of weighted norm inequalities for the conical square function Sα,L associated with an abstract operator L. We first establish two-weight inequalities including bump estimates, and Fefferman-Stein inequalities with arbitrary weights. We also present the local decay estimates using the extrapolation techniques, and the mixed weak type estimates corresponding to Sawyer's conjecture by means of a Coifman-Fefferman inequality. Beyond that, we consider other weak type estimates including the restricted weak-type (p,p) for Sα,L and the endpoint estimate for commutators of Sα,L. Finally, all the conclusions aforementioned can be applied to a number of square functions associated to L.

Application of capacities to space–time fractional dissipative equations I: regularity and the blow-up set

AbstractWe apply capacities to explore the space–time fractional dissipative equation: (0.1) $$ \begin{align} \left\{\begin{aligned} &\partial^{\beta}_{t}u(t,x)=-\nu(-\Delta)^{\alpha/2}u(t,x)+f(t,x),\quad (t,x)\in\mathbb R^{1+n}_{+},\\ &u(0,x)=\varphi(x),\ x\in\mathbb R^{n}, \end{aligned}\right. \end{align} $$ where $\alpha>n$ and $\beta \in (0,1)$ . In this paper, we focus on the regularity and the blow-up set of mild solutions to (0.1). First, we establish the Strichartz-type estimates for the homogeneous term $R_{\alpha ,\beta }(\varphi )$ and inhomogeneous term $G_{\alpha ,\beta }(g)$ , respectively. Second, we obtain some space–time estimates for $G_{\alpha ,\beta }(g).$ Based on these estimates, we prove that the continuity of $R_{\alpha ,\beta }(\varphi )(t,x)$ and the Hölder continuity of $G_{\alpha ,\beta }(g)(t,x)$ on $\mathbb {R}^{1+n}_+,$ which implies a Moser–Trudinger-type estimate for $G_{\alpha ,\beta }.$ Then, for a newly introduced $L^{q}_{t}L^p_{x}$ -capacity related to the space–time fractional dissipative operator $\partial ^{\beta }_{t}+(-\Delta )^{\alpha /2},$ we perform the geometric-measure-theoretic analysis and establish its basic properties. Especially, we estimate the capacity of fractional parabolic balls in $\mathbb {R}^{1+n}_+$ by using the Strichartz estimates and the Moser–Trudinger-type estimate for $G_{\alpha ,\beta }.$ A strong-type estimate of the $L^{q}_{t}L^p_{x}$ -capacity and an embedding of Lorentz spaces are also derived. Based on these results, especially the Strichartz-type estimates and the $L^{q}_{t}L^p_{x}$ -capacity of fractional parabolic balls, we deduce the size, i.e., the Hausdorff dimension, of the blow-up set of solutions to (0.1).

Quantitative matrix weighted estimates for certain singular integral operators

In this paper quantitative weighted matrix estimates for vector valued extensions of Lr′-Hörmander operators and rough singular integrals are studied. Strong type (p,p) estimates, endpoint estimates, and some new results on Coifman-Fefferman estimates assuming A∞ and Cp condition counterparts are provided. To prove the aforementioned estimates we rely upon some suitable convex body domination results that we settle as well in this paper.

Weighted $$L^{p}$$ estimates on the infinite rooted k-ary tree

In this paper sufficient conditions for weighted weak and strong type (p, p) estimates with \(p>1\) for the centered maximal function on the infinite rooted k-ary tree are provided. Here the line initated by the authors and Safe in Ombrosi et al. (Int Math Res Not IMRN 4:2736–2762, 2021) and providing a further extension of the use of techniques due to Naor and Tao in (J Funct Anal 259(3):731–779, 2010) is continued. The fact that the class of weights from the sufficient conditions is wider for \(L^p\) estimates than the one obtained in [16] is established as well. Some results highlighting the pathological nature of the weighted \(L^p\) theory in this setting are settled. It is shown that the \(A_p\) condition is no precise in this setting, since there exist weights such that the \(L^p\) boundedness holds but the \(A_p\) condition is not satisfied. It is also shown that the Sawyer type testing condition is not sufficient either for the strong type to hold and also that strong and weak type estimates are not equivalent in this setting. It will be shown as well that the one weight results can be extended to the two weight setting.

Weak and strong boundedness for p-adic fractional Hausdorff operator and its commutator

Abstract In this paper, the boundedness of the Hausdorff operator on weak central Morrey space is obtained. Furthermore, we investigate the weak bounds of the p-adic fractional Hausdorff operator on weighted p-adic weak Lebesgue spaces. We also obtain the sufficient condition of commutators of the p-adic fractional Hausdorff operator by taking symbol function from Lipschitz spaces. Moreover, strong type estimates for fractional Hausdorff operator and its commutator on weighted p-adic Lorentz spaces are also acquired.

Duality for Outer L^p_mu (ell ^r) Spaces and Relation to Tent Spaces

We study the outer L^p spaces introduced by Do and Thiele on sets endowed with a measure and an outer measure. We prove that, in the case of finite sets, for 1< p leqslant infty , 1 leqslant r < infty or p=r in { 1, infty }, the outer L^p_mu (ell ^r) quasi-norms are equivalent to norms up to multiplicative constants uniformly in the cardinality of the set. This is obtained by showing the expected duality properties between the corresponding outer L^p_mu (ell ^r) spaces uniformly in the cardinality of the set. Moreover, for p=1, 1 < r leqslant infty , we exhibit a counterexample to the uniformity in the cardinality of the finite set. We also show that in the upper half space setting the desired properties hold true in the full range 1 leqslant p,r leqslant infty . These results are obtained via greedy decompositions of functions in the outer L^p_mu (ell ^r) spaces. As a consequence, we establish the equivalence between the classical tent spaces T^p_r and the outer L^p_mu (ell ^r) spaces in the upper half space. Finally, we give a full classification of weak and strong type estimates for a class of embedding maps to the upper half space with a fractional scale factor for functions on mathbb {R}^d.

An extension of Calderón-Zygmund type singular integral with non-smooth kernel

In the present paper, we consider a kind of singular integralTf(x)=p.v.∫RnΩ(y)|y|n−βf(x−y)dy which can be viewed as an extension of the classical Calderón-Zygmund type singular integral. This kind of singular integral appears in the approximation of the surface quasi-geostrophic (SQG) equation from the generalized SQG equation. We establish an estimate of the singular integral in the Lq space for 1<q<∞ and a weak (1,1) type of the singular integral when 0<β<(q−1)nq without any smoothness assumed on Ω. Moreover, the bounds do not depend on β and the strong (q,q) type estimate and weak (1,1) type estimate of the Calderón-Zygmund type singular integral can be recovered when β→0 from our obtained estimates.

Weighted Estimates for Vector-Valued Intrinsic Square Functions and Commutators in the Morrey-Type Spaces

In this paper, the boundedness properties of vector-valued intrinsic square functions and their vector-valued commutators with \(BMO(\mathbb {R}^{n})\) functions are discussed. We first show the weighted strong-type and weak-type estimates of vector-valued intrinsic square functions in the Morrey-type spaces. Then, we obtain weighted strong-type estimates of vector-valued analogues of commutators in Morrey-type spaces. In the endpoint case, we establish the weighted weak \(L\log L\)-type estimates for these vector-valued commutators in the setting of weighted Lebesgue spaces. Furthermore, we prove weighted endpoint estimates of these commutator operators in Morrey-type spaces. In particular, we can obtain strong-type and endpoint estimates of vector-valued intrinsic square functions and their commutators in the weighted Morrey spaces and the generalized Morrey spaces.

Weak and strong type estimates for multilinear Calder\xf3n\u2013Zygmund operators on differential forms

In this paper, we define the multilinear Calderón–Zygmund operators on differential forms and prove the end-point weak type boundedness of the operators. Based on nonhomogeneous A-harmonic tensor, the Poincaré-type inequalities for multilinear Calderón–Zygmund operators on differential forms are obtained.

A noncommutative weak type (1,1) estimate for a square function from ergodic theory

In this paper, we investigate the boundedness of a square function operator from ergodic theory acting on noncommutative Lp-spaces. The main result is a weak type (1,1) estimate of this operator. We also show the (L∞,BMO) estimate, and thus all the strong type (Lp,Lp) estimates by interpolation. The main new difficulty lies in the fact that the kernel of this square function operator does not enjoy any regularity, while the Lipschitz regularity assumption is crucial in showing such endpoint estimates for the noncommutative Calderón-Zygmund singular integrals.

New weighted norm inequalities for multilinear Calder\xf3n\u2013Zygmund operators with kernels of Dini\u2019s type and their commutators

In this paper, we introduce certain classes of multilinear Calderón–Zygmund operators with kernels of Dini’s type. Applying the sharp method and A_{vec{p}}^{infty }(varphi ) functions, we first establish some weighted norm inequalities for multilinear Calderón–Zygmund operators with kernels of Dini’s type, including pointwise estimates, strong type, and weak endpoint estimates. Furthermore, similar weighted norm inequalities for commutators with mathrm{BMO}_{theta }(varphi ) functions are also obtained, but the weak endpoint estimate is of L({mathrm{log}}L) type.

Weighted inequalities for one-sided fractional minimal function

The purpose of this article is to present weighted norm inequalities for one-sided fractional minimal function. We prove weighted weak and strong type norm inequalities for the one-sided fractional minimal function on the real line. We construct two weight classes to ensure the necessary and sufficient condition for the weak and strong type estimates and establish an equivalence relation between the two weight classes.

Estimates for Fractional Integral Operators and Linear Commutators on Certain Weighted Amalgam Spaces

In this paper, we first introduce some new classes of weighted amalgam spaces. Then, we give the weighted strong-type and weak-type estimates for fractional integral operators Iγ on these new function spaces. Furthermore, the weighted strong-type estimate and endpoint estimate of linear commutators b,Iγ generated by b and Iγ are established as well. In addition, we are going to study related problems about two-weight, weak-type inequalities for Iγ and b,Iγ on the weighted amalgam spaces and give some results. Based on these results and pointwise domination, we can prove norm inequalities involving fractional maximal operator Mγ and generalized fractional integrals ℒ−γ/2 in the context of weighted amalgam spaces, where 0&lt;γ&lt;n and L is the infinitesimal generator of an analytic semigroup on L2Rn with Gaussian kernel bounds.

Limiting weak type behavior for multilinear fractional integrals

In this paper, we establish the limiting weak type behavior for multilinear fractional integrals Iα,A and Iα,A, which were introduced by Kenig and Stein (see the definition in section 1). Although Kenig and Stein proved that these two kinds of multilinear fractional operators have the same weak and strong type estimates, we provide that their limiting behaviors are totally different when p1=p2=⋯=pk=1. Specifically, motivated by Janakiraman’s result, we establish when fi≥0,1≤i≤k, there holds limλ→0λ|{x∈Rn:|Iα,k(f1,…,fk)(x)|>λ}|kn−αn=kα−knvnkn−αn∏i=1k‖fi‖L1(Rn), which extend the first author’s result about linear fractional integral to multilinear case. But for Iα,A, there holds limλ→0λ|{x∈Rn:|Iα,A(f1,…,fk)(x)|>λ}|kn−αn=0.Limiting behaviors for the other indexes are also presented and at last we set up limiting weak type behaviors for multilinear fractional integrals when λ→∞.

Weighted Morrey Spaces Related to Schrödinger Operators with Nonnegative Potentials and Fractional Integrals

Let ℒ=−Δ+V be a Schrödinger operator on ℝd, d≥3, where Δ is the Laplacian operator on ℝd, and the nonnegative potential V belongs to the reverse Hölder class RHs with s≥d/2. For given 0&lt;α&lt;d, the fractional integrals associated with the Schrödinger operator ℒ is defined by ℐα=ℒ−α/2. Suppose that b is a locally integrable function on ℝd and the commutator generated by b and ℐα is defined by b.ℐαfx=bx⋅ℐαfx−ℐαbfx. In this paper, we first introduce some kinds of weighted Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder class RHs with s≥d/2. Then, we will establish the boundedness properties of the fractional integrals ℐα on these new spaces. Furthermore, weighted strong-type estimate for the corresponding commutator b,ℐα in the framework of Morrey space is also obtained. The classes of weights, the classes of symbol functions, as well as weighted Morrey spaces discussed in this paper are larger than Ap,q, BMOℝd, and Lp,κμ,ν corresponding to the classical case (that is V≡0).

Weak and strong type estimates for the multilinear pseudo-differential operators

In this paper, we investigate the boundedness of the multilinear pseudo-differential operator Tσ. First, we establish the local exponential decay estimates for Tσ. In terms of the corresponding commutators Tσ,Σb, we obtain the local subexponential decay estimates. Secondly, we derive the weighted mixed weak type inequality for Tσ, which parallels Sawyer's conjecture for Calderón-Zygmund operators and covers the endpoint weighted inequalities. Last but not least, we present the sharp weighted estimates for Tσ and Tσ,Σb. It is worth mentioning that our results are totally new even in the linear case.

Variable martingale Hardy-Morrey spaces

In this paper, we introduce martingale Morrey spaces with variable exponents defined on a probability space. The weak-type and strong-type estimates for the Doob maximal operator on these spaces are established. We also construct the atomic decompositions for variable Hardy-Morrey spaces. As an application of the atomic decompositions, martingale inequalities among different Hardy-Morrey spaces are obtained. Further, we investigate the boundedness of fractional integral operators on martingale Hardy-Morrey spaces.

Weighted Morrey Spaces Related to Certain Nonnegative Potentials and Riesz Transforms

LetL=-Δ+Vbe a Schrödinger operator, whereΔis the Laplacian onRdand the nonnegative potentialVbelongs to the reverse Hölder classRHqforq≥d. The Riesz transform associated with the operatorL=-Δ+Vis denoted byR=∇(-Δ+V)-1/2and the dual Riesz transform is denoted byR⁎=(-Δ+V)-1/2∇. In this paper, we first introduce some kinds of weighted Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder classRHqforq≥d. Then we will establish the mapping properties of the operatorRand its adjointR⁎on these new spaces. Furthermore, the weighted strong-type estimate and weighted endpoint estimate for the corresponding commutators[b,R]and[b,R⁎]are also obtained. The classes of weights, classes of symbol functions, and weighted Morrey spaces discussed in this paper are larger thanAp,BMO(Rd), andLp,κ(w)corresponding to the classical Riesz transforms (V≡0).

Weak and Strong Boundedness for p-ADIC Fractional Hausdorff Operator and Its Commutator

In this paper, boundedness of Hausdorff operator on weak central Morrey space is obtained. Furthermore, we investigate the weak bounds of p- adic fractional Hausdorff Operator on weighted p-adic weak Lebesgue Space. We also obtain the sufficient condition of commutators of p-adic fractional Hausdorff Operator by taking symbol function from Lipschitz space. Moreover, strong type estimates for fractional Hausdorff Operator and its commutator on weighted p-adic Lorentz space are also acquired.

Convolution estimates for measures on some complex curves

We consider the convolution operator for a measure supported on complex curves. The measure which we consider here is an analogue of the affine arclength measure for real curves. By modifying a combinatorial argument called the band structure argument, we prove the (nearly) optimal Lorentz space estimates. This includes the optimal strong type estimates as special cases. The complex curves we consider here are the ones considered for the Fourier restriction estimates for complex curves in \cite{BH}.