Extrapolation to weighted Morrey spaces with variable exponents and applications

Mapping properties of operator-valued pseudo-differential operators

Weak type estimates for genuine Calderón–Zygmund operators are established on the local Morrey spaces associated with ball quasi-Banach function spaces by two different methods. Above all, we obtain weak type estimates for the operator on the local weak Morrey spaces with variable exponents.

We obtain the boundedness of Calderón–Zygmund operators on the entire Hardy–Morrey spaces with variable exponents. We obtain this result by refining the extrapolation theory. This refined extrapolation theory also gives the boundedness of Fourier multipliers, Bochner–Riesz means, and parametric Marcinkiewicz integrals on Hardy–Morrey spaces with variable exponents.

In this paper, we consider the 3-D generalized micropolar fluid system in critical Fourier-Besov- Morrey spaces with variable exponent. Using the Littlewood-Paley theory and Banach fixed point theorem we establish the global existence result with the small initial data belonging to F? 4?2??3/p(?)p(?),h(?),q (R3).

In this paper, we establish the boundedness of the Calderón operator on local Morrey spaces with variable exponents. We obtain our result by extending the extrapolation theory of Rubio de Francia to the local Morrey spaces with variable exponents. The exponent functions of the local Morrey spaces with the exponent functions are only required to satisfy the log-Hölder continuity assumption at the origin and infinity only. As special cases of the main result, we have Hardy’s inequalities, the Hilbert inequalities and the boundedness of the Riemann–Liouville and Weyl averaging operators on local Morrey spaces with variable exponents.

We investigate the direct and inverse theorems for trigonometric polynomials in the Morrey space $${{\mathcal {M}}}^{p(\cdot ),\lambda (\cdot )}$$ with variable exponents. For this space, we obtain estimates of the K-functional in terms of the modulus of smoothness and the Bernstein type inequality for trigonometric polynomials.

<p>This paper intends to demonstrate the boundedness of the fractional bilinear Hardy operator and its adjoint on the $ \lambda $-central Morrey space with variable exponents. Analogous outcomes for their commutators are derived when the symbol functions are elements of the $ \lambda $-central bounded mean oscillation ($ \lambda $-central BMO) space.</p>

In this paper, using the atomic decomposition of the Herz-Morrey-Hardy spaces with variable exponent, the wavelet characterization by means of a local version of the discrete tent spaces with variable exponent is established. As an application, the boundedness of the fractional integral operators from variable exponent Herz-Morrey-Hardy spaces into variable exponent Herz-Morrey spaces is obtained.

This paper extends the celebrated Stein–Weiss inequalities to the Herz spaces with variable exponents. It is an extension of the classical Stein–Weiss inequalities and the Stein–Weiss inequalities on Lebesgue spaces with variable exponents. The Stein–Weiss inequalities to the Herz spaces with variable exponents are presented as the boundedness of the fractional integral operators on the power weighted Herz spaces with variable exponents.

"In this paper we consider weighted Morrey spaces $L_{p,\lambda,|\cdot|^{\gamma}}(\Rn).$ We prove the Hardy-Littlewood-Stein-Weiss type $L_{p,\lambda,|\cdot|^{\gamma}}(\Rn)$ to $L_{q,\lambda,|\cdot|^{\mu}}(\Rn)$ theorems for Riesz potential $I^{\al}$ and its commutators $[b,I^{\alpha}]$ and $|b,I^{\alpha}|$, where $0<\alpha<n$, $0\leq\lambda <n-\alpha$, $1< p<\frac {n-\lambda}{\alpha}$, $-n+\lambda\le\gamma<n(p-1) +\lambda$, $\mu=\frac{q\gamma}{p}$, $\frac 1p-\frac 1q = \frac \alpha {n-\lambda}$, $b\in BMO(\Rn).$ As a result of these we obtain the conditions for the boundedness of the commutator $|b,I^{\al}|$ from Besov-Morrey spaces $B_{p,\theta,\lambda,|\cdot|^{\gamma}}^s (\Rn)$ to $B_{q,\theta,\lambda,|\cdot|^{\mu}}^s (\Rn).$ Furthermore, we consider the Schr\""{o}dinger operator $-\Delta + V$ on $\Rn$ and obtain weighted Morrey $L_{p,\lambda,|\cdot|^{\gamma}}(\Rn)$ estimates for the operators $V^{s} (-\Delta+V)^{-\beta}$ and $V^{s} \nabla (-\Delta+V)^{-\beta}.$ Finally we apply our results to various operators which are estimated from above by Riesz potentials."

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<α<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).

‎In this paper‎, ‎using the atomic theory of the Herz-type Hardy spaces‎ ‎with variable exponent‎, ‎we give their wavelet characterization by‎ ‎means of some discrete tent spaces with variable exponent at the‎ ‎origin‎.

We study some multilinear operators with rough kernels. For the multilinear fractional integral operatorsTΩ,αAand the multilinear fractional maximal integral operatorsMΩ,αA, we obtain their boundedness on weighted Morrey spaces with two weightsLp,κ(u,v)whenDγA∈Λ˙β (|γ|=m-1)orDγA∈BMO (|γ|=m-1). For the multilinear singular integral operatorsTΩAand the multilinear maximal singular integral operatorsMΩA, we show they are bounded on weighted Morrey spaces with two weightsLp,κ(u,v)ifDγA∈Λ˙β (|γ|=m-1)and bounded on weighted Morrey spaces with one weightLp,κ(w)ifDγA∈BMO (|γ|=m-1)form=1,2.

We study the weighted boundedness of the Cauchy singular integral operator S Γ in Morrey spaces L p , λ ( Γ ) on curves satisfying the arc-chord condition, for a class of “radial type” almost monotonic weights. The non-weighted boundedness is shown to hold on an arbitrary Carleson curve. We show that the weighted boundedness is reduced to the boundedness of weighted Hardy operators in Morrey spaces L p , λ ( 0 , ℓ ) , ℓ > 0 . We find conditions for weighted Hardy operators to be bounded in Morrey spaces. To cover the case of curves we also extend the boundedness of the Hardy–Littlewood maximal operator in Morrey spaces, known in the Euclidean setting, to the case of Carleson curves.

Via the boundedness of intrinsic g-functions from the Hardy spaces with variable exponent, ℋp(·)(ℝn), into Lebesgue spaces with variable exponent, Lp(·)(ℝn), and establishing some estimates on a discrete Littlewood-Paley g-function and a Peetre-type maximal function, we obtain several equivalent characterizations of ℋp(·)(ℝn) in terms of wavelets, which extend the wavelet characterizations of the classical Hardy spaces. The main ingredients are that, we overcome the difficulties of the quasi-norms of ℋp(·)(ℝn) by elaborately using an observation and the Fefferman-Stein vector-valued maximal inequality on Lp(·)(ℝn), and also overcome the difficulty of the failure of q = 2 in the atomic decomposition of ℋp(·)(ℝn) by a known idea.