31,772 research outputs found
H\"ormander Type Functional Calculus and Square Function Estimates
We investigate H\"ormander spectral multiplier theorems as they hold on for many self-adjoint elliptic differential
operators including the standard Laplacian on A strengthened
matricial extension is considered, which coincides with a completely bounded
map between operator spaces in the case that is a Hilbert space. We show
that the validity of the matricial H\"ormander theorem can be characterized in
terms of square function estimates for imaginary powers , for
resolvents and for the analytic semigroup We
deduce H\"ormander spectral multiplier theorems for semigroups satisfying
generalized Gaussian estimates
Endpoint multiplier theorems of Marcinkiewicz type
We establish sharp (H^1, L^{1,q}) and local (L \log^r L, L^{1,q}) mapping
properties for rough one-dimensional multipliers. In particular, we show that
the multipliers in the Marcinkiewicz multiplier theorem map H^1 to L^{1,\infty}
and L \log^{1/2} L to L^{1,\infty}, and that these estimates are sharp.Comment: 28 pages, no figures, submitted to Revista Mat. Ibe
Lp Fourier multipliers on compact Lie groups
In this paper we prove Lp multiplier theorems for invariant and non-invariant
operators on compact Lie groups in the spirit of the well-known
Hormander-Mikhlin theorem on Rn and its variants on tori Tn. We also give
applications to a-priori estimates for non-hypoelliptic operators. Already in
the case of tori we get an interesting refinement of the classical multiplier
theorem.Comment: 22 pages; minor correction
Fourier multiplier theorems involving type and cotype
In this paper we develop the theory of Fourier multiplier operators
, for Banach spaces
and , and an operator-valued symbol. The case has been studied
extensively since the 1980's, but far less is known for . In the scalar
setting one can deduce results for from the case . However, in the
vector-valued setting this leads to restrictions both on the smoothness of the
multiplier and on the class of Banach spaces. For example, one often needs that
and are UMD spaces and that satisfies a smoothness condition. We
show that for other geometric conditions on and , such as the
notions of type and cotype, can be used to study Fourier multipliers. Moreover,
we obtain boundedness results for without any smoothness properties of
. Under smoothness conditions the boundedness results can be extrapolated to
other values of and as long as remains
constant.Comment: Revised version, to appear in Journal of Fourier Analysis and
Applications. 31 pages. The results on Besov spaces and the proof of the
extrapolation result have been moved to arXiv:1606.0327
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