Endpoint estimates for a trilinear pseudo-differential operator with flag symbols

In this paper, we establish the endpoint estimate ( 0 < p ≤ 1 {0<p\leq 1} ) for a trilinear pseudo-differential operator, where the symbol involved is given by the product of two standard symbols from the bilinear Hörmander class B ⁢ S 1 , 0 0 {BS^{0}_{1,0}} . The study of this operator is motivated from the L p {L^{p}} ( 1 < p < ∞ {1<p<\infty} ) estimates for the trilinear Fourier multiplier operator with flag singularities considered in [C. Muscalu, Paraproducts with flag singularities. I. A case study, Rev. Mat. Iberoam. 23 2007, 2, 705–742] and Hardy space estimates in [A. Miyachi and N. Tomita, Estimates for trilinear flag paraproducts on L ∞ L^{\infty} and Hardy spaces, Math. Z. 282 2016, 1–2, 577–613], and the L p {L^{p}} ( 1 < p < ∞ {1<p<\infty} ) estimates for the trilinear pseudo-differential operator with flag symbols in [G. Lu and L. Zhang, L p L^{p} -estimates for a trilinear pseudo-differential operator with flag symbols, Indiana Univ. Math. J. 66 2017, 3, 877–900]. More precisely, we will show that the trilinear pseudo-differential operator with flag symbols defined in (1.3) maps from the product of local Hardy spaces to the Lebesgue space, i.e., h p 1 × h p 2 × h p 3 → L p {h^{p_{1}}\times h^{p_{2}}\times h^{p_{3}}\rightarrow L^{p}} with 1 p 1 + 1 p 2 + 1 p 3 = 1 p {\frac{1}{p_{1}}+\frac{1}{p_{2}}+\frac{1}{p_{3}}=\frac{1}{p}} with 0 < p < ∞ {0<p<\infty} (see Theorem 1.6 and Theorem 1.7).

Mean oscillation and boundedness of multilinear operator related to multiplier operator

In this paper, the boundedness of certain multilinear operator related to the multiplier operator from Lebesgue spaces to Orlicz spaces is obtained.

A Note on the $$L^p$$ Integrability of a Class of Bochner–Riesz Kernels

For a general compact variety $$\Gamma $$ Γ of arbitrary codimension, one can consider the $$L^p$$ L p mapping properties of the Bochner–Riesz multiplier $$\begin{aligned} m_{\Gamma , \alpha }(\zeta ) \ = \ \mathrm{dist}(\zeta , \Gamma )^{\alpha } \phi (\zeta ) \end{aligned}$$ m Γ , α ( ζ ) = dist ( ζ , Γ ) α ϕ ( ζ ) where $$\alpha > 0$$ α > 0 and $$\phi $$ ϕ is an appropriate smooth cutoff function. Even for the sphere $$\Gamma = {{\mathbb {S}}}^{N-1}$$ Γ = S N - 1 , the exact $$L^p$$ L p boundedness range remains a central open problem in Euclidean harmonic analysis. In this paper we consider the $$L^p$$ L p integrability of the Bochner–Riesz convolution kernel for a particular class of varieties (of any codimension). For a subclass of these varieties the range of $$L^p$$ L p integrability of the kernels differs substantially from the $$L^p$$ L p boundedness range of the corresponding Bochner–Riesz multiplier operator.

The convolution of finite number of analytic functions

We investigate various results associated with the convolution of finite number of analytic functions involving a certain multiplier operator (defined below). Some useful consequences including a result related to the zeta function are also mentioned.

Duals and multipliers of controlled frames in Hilbert spaces

In this paper, we introduce and characterize controlled dual frames in Hilbert spaces. We also investigate the relation between bounds of controlled frames and their related frames. Then, we define the concept of approximate duality for controlled frames in Hilbert spaces. Next, we introduce multiplier operators of controlled frames in Hilbert spaces and investigate some of their properties. Finally, we show that the inverse of a controlled multiplier operator is also a controlled multiplier operator under some mild conditions.

ON THE BILINEAR SQUARE FOURIER MULTIPLIER OPERATORS ASSOCIATED WITH FUNCTION

This paper will be devoted to study a class of bilinear square-function Fourier multiplier operator associated with a symbol $m$ defined by $$\begin{eqnarray}\displaystyle & & \displaystyle \mathfrak{T}_{\unicode[STIX]{x1D706},m}(f_{1},f_{2})(x)\nonumber\\ \displaystyle & & \displaystyle \quad =\Big(\iint _{\mathbb{R}_{+}^{n+1}}\Big(\frac{t}{|x-z|+t}\Big)^{n\unicode[STIX]{x1D706}}\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,\bigg|\int _{(\mathbb{R}^{n})^{2}}e^{2\unicode[STIX]{x1D70B}ix\cdot (\unicode[STIX]{x1D709}_{1}+\unicode[STIX]{x1D709}_{2})}m(t\unicode[STIX]{x1D709}_{1},t\unicode[STIX]{x1D709}_{2})\hat{f}_{1}(\unicode[STIX]{x1D709}_{1})\hat{f}_{2}(\unicode[STIX]{x1D709}_{2})\,d\unicode[STIX]{x1D709}_{1}\,d\unicode[STIX]{x1D709}_{2}\bigg|^{2}\frac{dz\,dt}{t^{n+1}}\Big)^{1/2}.\nonumber\end{eqnarray}$$ A basic fact about $\mathfrak{T}_{\unicode[STIX]{x1D706},m}$ is that it is closely associated with the multilinear Littlewood–Paley $g_{\unicode[STIX]{x1D706}}^{\ast }$ function. In this paper we first investigate the boundedness of $\mathfrak{T}_{\unicode[STIX]{x1D706},m}$ on products of weighted Lebesgue spaces. Then, the weighted endpoint $L\log L$ type estimate and strong estimate for the commutators of $\mathfrak{T}_{\unicode[STIX]{x1D706},m}$ will be demonstrated.

A Multiplier Theorem on Anisotropic Hardy Spaces

We present a multiplier theorem on anisotropic Hardy spaces. When m satisfies the anisotropic, pointwise Mihlin condition, we obtain boundedness of the multiplier operator Tm: → , for the range of p that depends on the eccentricities of the dilation A and the level of regularity of a multiplier symbol m. This extends the classical multiplier theorem of Taibleson andWeiss.

A New Array Multiplier Using an Optimized Carry Network and Dynamic CMOS Technology

In this paper, a new multiplier using array architecture and a fast carry network tree is presented which uses dynamic CMOS technology. Different reforms are performed in multiplier architecture. In the first step of multiplier operator, a novel radix-16 modified Booth encoder is presented which reduces the number of partial products efficiently. In this research, we present a new algorithm for partial product reduction in multiplication operations. The algorithm is based on the implementation of compressor elements by means of carry network. The structure of these compressors into reduction trees takes advantage of the modified Wallace tree for integration of adder cells and provides an alternative to conventional operator methods. We show several reduction techniques that illustrate the proposed method and describe carry-skip examples that combine dynamic CMOS with classic conventional compressors in order to modify each scheme. In network multiplier, a novel low power high-speed adder cell is presented which uses 14 transistors in its structure. Critical path is minimized to reduce latency in whole operator architecture. Final adder of multiplier uses an optimized carry hybrid adder. The presented final adder network uses dynamic CMOS technology. It sums two final operands in a very efficient way, which has significant effect in operator structure. Presented multiplier reduces latency by 12%, decreases transistor count by 8% and modifies noise problem in an efficient way in comparison with other structures.