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 the s-nuclearity, 0 < s ≤ 1, of periodic and discrete pseudo-differential operators. To accomplish this, we classify those s-nuclear multilinear integral operators on arbitrary Lebesgue spaces defined on σ-finite measures spaces. We also study similar properties for periodic Fourier integral operators. Finally, we present some applications of our study to deduce the periodic Kato-Ponce inequality and to examine the s-nuclearity of multilinear Bessel potentials as well as the s-nuclearity of periodic Fourier integral operators admitting suitable types of singularities.MSC 2010. Primary 58J40.; Secondary 47B10, 47G30, 35S30. ContentsDate: Received: xxxxxx; Revised: yyyyyy; Accepted: zzzzzz. 2010 Mathematics Subject Classification. 58J40.; Secondary 47B10, 47G30, 35S30 .
In this note we announce our investigation on the L p properties for periodic and discrete multilinear pseudo-differential operators. First, we review the periodic analysis of multilinear pseudo-differential operators by showing classical multilinear Fourier multipliers theorems (proved by Coifman and Meyer, Tomita, Miyachi, Fujita, Grafakos, 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. The s-nuclearity, 0 < s ≤ 1, for the discrete and periodic multilinear pseudo-differential operators will be investigated. To do so, we classify those s-nuclear, 0 < s ≤ 1, multilinear integral operators on arbitrary Lebesgue spaces defined on σ-finite measures spaces. Finally, we present some applications of our analysis to deduce the periodic Kato-Ponce inequality and to examine the s-nuclearity of multilinear Bessel potentials as well as the s-nuclearity of periodic Fourier integral operators admitting suitable types of singularities.
For a locally compact hypergroup K and a Young function ϕ, we study the Orlicz space L ϕ (K) and provide a sufficient condition for L ϕ (K) to be an algebra under convolution of functions. We show that a closed subspace of L ϕ (K) is a left ideal if and only if it is left translation invariant. We apply the basic theory developed here to characterize the space of multipliers of the Morse-Transue space M ϕ (K). We also investigate the multipliers of L ϕ (S, πK ), where S is the support of the Plancherel measure πK associated to a commutative hypergroup K.
Given a compact (Hausdorff) group G and a closed subgroup H of G, in this paper we present symbolic criteria for pseudo-differential operators on compact homogeneous space G/H characterizing the Schatten-von Neumann classes Sr(L 2 (G/H)) for all 0 < r ≤ ∞. We go on to provide a symbolic characterization for r-nuclear, 0 < r ≤ 1, pseudo-differential operators on L p (G/H)-space with applications to adjoint, product and trace formulae. The criteria here are given in terms of the concept of matrix-valued symbols defined on noncommutative analogue of phase space G/H × G/H. Finally, we present applications of aforementioned results in the context of heat kernels. be in Schatten-von Neumann class S r of operators on L 2 (G/H) for 0 < r ≤ ∞; (2) to find criteria for pseudo-differential operators from L p1 (G/H) into L p2 (G/H) to be r-nuclear, 0 < r ≤ 1, for 1 ≤ p 1 , p 2 < ∞; and (3) applications to find a trace formula and to provide criteria for heat kernel to be nuclear on L p (G/H). In order to do this, we will use the global quantization developed for compact homogeneous spaces as a non-commutative analogue of the Kohn-Nirenberg quantization of operators on R n .Recently, several researchers started a extensive research for finding the criteria for Schatten classes and r-nuclear operators in terms of symbols with lower regularity [2,12,15,40,41]. Ruzhansky and Delgado [12,14] successfully drop the regularity condition at least in their setting using the matrix-valued symbols instead of standard Kohn-Nirenberg quantization. Inspired by the work of Delgado and Ruzhansky, we present symbolic criteria for pseudo-differential operators on G/H to be Schatten class using the matrix-valued symbols defined on G/H × G/H, a noncommutative analogue of phase space. It is well known that in the setting of Hilbert spaces the class of r-nuclear operators agrees with the Schatten-von Neumann ideal of order r [35]. In general, for trace class operators on Hilbert spaces, the trace of an operator given by integration of its integral kernel over the diagonal is equal to the sum of its eigenvalues. However, this property fails in Banach spaces. The importance of r-nuclear operator lies in the work of Grothendieck, who proved that for 2/3-nuclear operators, the trace in Banach spaces agrees with the sum of all the eigenvalues with multiplicities counted. Therefore, the notion of r-nuclear operators becomes useful. Now, the question of finding good criteria for ensuring the r-nuclearity of operators arises but this has to be formulated in terms different from those on Hilbert spaces and has to take into account the impossibility of certain kernel formulations in view of Carleman's example [6] (also see [12]). In view of this, we will establish conditions imposed on symbols instead of kernels ensuring the r-nuclearity of the corresponding operators.The initiative of finding the necessary and sufficient conditions for a pseudo-differential operator defined on a group to be r-nuclear has been started by Delgado and Wong [16]. The main ingr...
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