We give explicit necessary and sufficient conditions for the boundedness of the general second order differential operator L with real- or complex-valued distributional coefficients acting from the Sobolev space W^{1,2}(R^n) to its dual W^{-1,2}(R^n). This enables us to obtain analytic criteria for the fundamental notions of relative form boundedness, compactness, and infinitesimal form boundedness of L with respect to the Laplacian on L^2(R^n).
In particular, we establish a complete characterization of the form boundedness of the Schroedinger operator (i \nabla + a)^2 + q with magnetic vector potential a \in L^2_{loc} and q \in D'(\R^n).
We prove that pseudo-differential operators with symbols in the class $S_{1,δ}^0$ ($0<δ<1$) are not always bounded on the modulation space $M^{p,q}$ ($q\neq2$).
In this paper, we study the boundedness of pseudo-differential operators with symbols in $S_{ρ,δ}^m$ on the modulation spaces $M^{p,q}$. We discuss the order $m$ for the boundedness $\mathrm{Op}(S_{ρ,δ}^m) \subset \calL(M^{p,q}(\R^n))$ to be true. We also prove the existence of a Calderón-Zygmund operator which is not bounded on the modulation space $M^{p,q}$ with $q \neq 2$. This unboundedness is still true even if we assume a generalized T(1) condition. These results are induced by the unboundedness of pseudo-differential operators on $M^{p,q}$ whose symbols are of the class $S_{1,δ}^0$ with…
In this paper we study the boundedness of global pseudo-differential operators on smooth manifolds. By using the notion of global symbol we extend a classical condition of Hörmander type to guarantee the $L^p$-$L^q$-boundedness of global operators. First we investigate $L^p$-boundedness of pseudo-differential operators in view of the Hörmander-Mihlin condition. We also prove $L^\infty$-$BMO$ estimates for pseudo-differential operators. Later, we concentrate our investigation to settle $L^p$-$L^q$ boundedness of the Fourier multipliers and pseudo-differential operators for the range $1<p \leq 2…
We give a criterion for the global boundedness of integral operators which are known to be locally bounded. As an application, we discuss the global $L^p$-boundedness for a class of Fourier integral operators. While the local $L^p$-boundedness of Fourier integral operators is known from the work of Seeger, Sogge and Stein, not so many results are available for the global boundedness on $L^p({\mathbb R}^n)$. We give several natural sufficient conditions for them.
In this paper, the sharp maximal theorem is generalized to mixed-norm ball Banach function spaces, which is defined as Definition 2.7. As an application, we give a characterization of BMO via the boundedness of commutators of fractional integral operators on mixed-norm Lebesgue spaces. Moreover, the characterization of homogeneous Lipschitz space is also given by the boundedness of commutators of fractional integral operators on mixed-norm Lebesgue spaces. Finally, two applications of Corollary 6.4 are given.
In this paper, we provide the boundedness property of the Riesz transforms associated to the Schrödinger operator $\mathcal{L}=-Δ+ \mathbf{V}$ in a new weighted Morrey space which is the generalized version of many previous Morrey type spaces. The additional potential $\V$ considered in this paper is a non-negative function satisfying the suitable reverse Hölder's inequality. Our results are new and general in many cases of problems. As an application of the boundedness property of these singular integral operators, we obtain some regularity results of solutions to Schrödinger equations in the…
In this paper we investigate the $L^p$-boundedness of certain classes of periodic pseudo-differential operators. The operators considered arise from the study of symbols on $\mathbb{T}^n\times\mathbb{Z}^n$ with limited regularity.
We investigate properties of pseudodifferential operators on $L^2$ space on manifold with ends including asymptotically conical or hyperbolic ends. Our pseudodifferential operators are a generalization of the canonical quantization which naturally appears in the quantum mechanics on curved spaces. We prove a Calderón-Vaillancourt type theorem for our pseudodifferential operators and discuss a construction of parametrix of elliptic differential operators on manifolds with ends.
In this work we show endpoint boundedness properties of pseudo-differential operators of type $(ρ,ρ)$, $0<ρ<1$, on Triebel-Lizorkin and Besov spaces. Our results are sharp and they also cover operators defined by compound symbols.
In this work we establish sharp boundedness results for pseudo-differential operators corresponding to $a\in\mathcal{S}_{0,0}^{m}$ on Triebel-Lizorkin spaces $F_p^{s,q}$ and Besov spaces $B_p^{s,q}$.
In this article we consider a class of state-dependent delay differential equations which is modelling the dynamics of the number of adult trees in forests. We prove the boundedness of solutions for a single species model as well as a two-species model.
In this article, we begin a systematic study of the boundedness and the nuclearity properties of multilinear periodic pseudo-differential operators and multilinear discrete pseudo-differential operators on $L^p$-spaces. First, we prove analogues of known multilinear Fourier multipliers theorems (proved by Coifman and Meyer, Grafakos, Tomita, Torres, Kenig, Stein, Fujita, Tao, etc.) in the context of periodic and discrete multilinear pseudo-differential operators. For this, we use the periodic analysis of pseudo-differential operators developed by Ruzhansky and Turunen. Later, we investigate th…
We survey a few results on the boundedness of operators arising from the Weyl-Pedersen calculus associated with irreducible representations of nilpotent Lie groups.
We study the boundedness of rough Fourier integral and pseudodifferential operators, defined by general rough Hörmander class amplitudes, on Banach and quasi-Banach $L^p$ spaces. Thereafter we apply the aforementioned boundedness in order to improve on some of the existing boundedness results for Hörmander class bilinear pseudodifferential operators and certain classes of bilinear (as well as multilinear) Fourier integral operators. For these classes of amplitudes, the boundedness of the aforementioned Fourier integral operators turn out to be sharp. Furthermore we also obtain results for roug…
In this paper, we prove the boundedness of Bessel-Riesz operators on generalized Morrey spaces. The proof uses the usual dyadic decomposition, a Hedberg-type inequality for the operators, and the boundedness of Hardy-Littlewood maximal operator. Our results reveal that the norm of the operators is dominated by the norm of the kernels.
We study the global boundedness of bilinear and multilinear Fourier integral operators on Banach and quasi-Banach $L^p$ spaces, where the amplitudes of the operators are smooth or rough in the spatial variables. The results are obtained by proving suitable global boundedness of rough linear Fourier integral operators with amplitudes that behave as $L^{p}$ functions in the spatial variables. The bilinear and multilinear boundedness estimates are proven by using either an iteration procedure or decomposition of the amplitudes, and thereafter applying our global results for linear Fourier integra…
We give a simple proof of the boundedness of the fractional maximal operator providing in this way an alternative approach to the one given by C. Capone, D. Cruz Uribe and A. Fiorenza in \cite{CCUF}.
In this paper we investigate $L^p$ and Sobolev boundedness of a certain class of pseudodifferential operators with non-regular symbols. We employ regularisation methods, namely convolution with a net of mollifiers $(ρ_\eps)_\eps$, and we study the corresponding net of pseudodifferential operators providing $L^p$ and Sobolev estimates which relate the parameter $\eps$ with the non-regularity of the symbol.
In the context of variable exponent Lebesgue spaces equipped with a lower Ahlfors measure we obtain weighted norm inequalities over bounded domains for the centered fractional maximal function and the fractional integral operator.
We first consider two types of localizations of singular integral operators of convolution type, and show, under mild decay and smoothness conditions on the auxiliary functions, that their boundedness on the local Hardy space $h^1(\mathbb{R}^n)$ is equivalent. We then study the boundedness on $h^1(\mathbb{R}^n)$ of the commutator $[b,T]$ of an inhomogeneous singular integral operator with $b$ in $bmo(\mathbb{R}^n)$, the nonhomogeneous space of functions of bounded mean oscillation. We define local analogues of the atomic space $H^1_b(\mathbb{R}^n)$ introduced by Pérez in the case of the homoge…
In this article, the authors establish a general (two-weight) boundedness criterion for a pair of functions, $(F,f)$, on $\mathbb{R}^n$ in the scale of weighted Lebesgue spaces, weighted Lorentz spaces, (Lorentz--)Morrey spaces, and variable Lebesgue spaces. As applications, the authors give a unified approach to prove the (two-weight) boundedness of Calderón--Zygmund operators, Littlewood--Paley $g$-functions, Lusin area functions, Littlewood--Paley $g^\ast_λ$-functions, and fractional integral operators, in the aforementioned function spaces. Moreover, via applying the above (two-weight) bou…
We give another proof of the $L^p$ boundedness of scattering wave operators, at the low frequency part of the data. The proof also allows the control of the commutator of multiplication by $|x|$ with the wave operator in $L^p$. The method we develop here is geared to proving similar results for \emph{time dependent potentials}, complementing previous work focused on the high frequency part.
In this work we present necessary cancellation conditions for the continuity of linear operators in $h^p(\mathbb{R}^n)$, $0<p\leq 1$, that map atoms into pseudo-molecules. Our necessary condition, expressed in terms of the $T^{\ast}$ condition, is the same as the one recently proved sufficient in [3], thus providing a necessary and sufficient cancellation condition for the boundedness of inhomogeneous Calderón--Zygmund type operators
We prove the global $L^2 \times L^2 \to L^1$ boundedness of bilinear Fourier integral operators with amplitudes in $S^0_{1,0} (n,2)$. To achieve this, we require that the phase function can be written as $(x,ξ,η) \mapsto \phase_1(x,ξ) + \phase_2(x,η)$ where each $\phase_j$ belongs to the class $Φ^2$ and satisfies the strong non-degeneracy condition. This result extends that of R. Coifman and Y. Meyer regarding pseudodifferential operators to the case of Fourier integral operators.
We establish global regularity of multilinear Fourier integral operators that are associated to nonlinear wave equations on product of $L^p$ spaces by proving endpoint boundedness on suitable products spaces containing combinations of the local Hardy space, the local BMO and the $L^2$ spaces.
We generalize the recent result of Erdo{\u g}an, Goldberg and Green on the $L^p$-boundedness of wave operators for two dimensional Schrödinger operators and prove that they are bounded in $L^p(\R^2)$ for all $1<p<\infty$ if and only if the Schrödinger operator possesses no $p$-wave threshold resonances, viz. Schrödinger equation $(-\lap + V(x))u(x)=0$ possesses no solutions which satisfy $u(x)= (a_1x_1+a_2 x_2)|x|^{-2}+ o(|x|^{-1})$ as $|x|\to \infty$ for an $(a_1, a_2) \in \R^2\setminus \{(0,0)\}$ and, otherwise, they are bounded in $L^p(\R^2)$ for $1<p\leq 2$ and unbounded for $2<p<\infty$. …
