178,023 research outputs found
Some Grüss Type Inequalities for Vector-Valued Functions in Banach Spaces and Applications
No full textSome Grüss type inequalities for the Bochner integral of vectorvalued
functions in real or complex Banach spaces are given. Applications in
connection to the Heisenberg inequality for functions with values in Hilbert
spaces are also pointed out
Hermite-Hadamard's Inequality and the p-HH-Norm on the Cartesian Product of Two Copies of a Normed Space
No full textThe Cartesian product of two copies of a normed space is naturally equipped with
the well-known p-norm. In this paper, another notion of norm is introduced, and will be
called the p-HH-norm. This norm is an extension of the generalised logarithmic mean and is
connected to the p-norm by the Hermite-Hadamard's inequality. The Cartesian product space
(with respect to both norms) is complete, when the (original) normed space is. A proof for the
completeness of the p-HH-norm via Ostrowski's inequality is provided. This space is embedded
as a subspace of the well-known Lebesgue-Bochner function space (as a closed subspace, when
the norm is a Banach norm). Consequently, its geometrical properties are inherited from those
of Lebesgue-Bochner space. An explicit expression of the superior (inferior) semi-inner product
associated to both norms is considered and used to provide alternative proofs for the smoothness
and reflexivity of this space
Methods for the analysis of oscillatory integrals and Bochner-Riesz operators
No full textFor a smooth surface Γ of arbitrary codimension, one can consider the Lp mapping properties of the Bochner-Riesz multiplier
m(ζ) = dist(ζ,Γ)^α φ(ζ),
where α > 0 and φ is an appropriate smooth cutoff function. Even for the sphere, the exact Lp boundedness range remains a central open problem in Euclidean harmonic analysis. We consider the Lp integrability of the Bochner-Riesz convolution kernel for a particular class of surfaces (of any codimension). For a subclass of these surfaces the range of Lp integrability of the kernels differs substantially from the Lp boundedness range of the corresponding Bochner-Riesz multiplier operator. Extending work of Mockenhoupt, we then establish a range of operator bounds, which are sharp in the α exponent, under the assumption of an appropriate L2 restriction estimate. Hickman and Wright established sharp oscillatory integral estimates, associated with a particular class of surfaces, and derived restriction estimates. We extend this work to certain curves of standard type and corresponding surfaces of revolution. These surfaces are discussed as an explicit class for which we have Lp → Lp boundedness of the corresponding Bochner-Riesz operators.
Understanding the structure of the roots of real polynomials is important in obtaining stable bounds for oscillatory integrals with polynomial phases. For real polynomials with exponents in some fixed set,
Ψ(t)=x+y1 t^{k1} +...+yL t^{kL},
we analyse the different possible root structures that can occur as the coefficients vary. We first establish a stratification of roots into tiers containing roots of comparable sizes. We then show that at most L non-zero roots can cluster about a point. Supposing additional restrictions on the coefficients, we derive structural refinements. These structural results extend work of Kowalski and Wright and provide a characteristic picture of root structure at coarse scales. As an application, these results are used to recover the sharp oscillatory integral estimates of Hickman and Wright, using bounds for oscillatory integrals of Phong and Stein
Bochner-Riesz means for critical magnetic Schr\"odinger operators in
No full textWe study -boundedness of the Bochner-Riesz means for critical magnetic
Schr\"odinger operators in , which
involve the physcial Aharonov-Bohm potential. We show that for and , the Bochner-Riesz operator
of order is bounded on
if and only if . The new ingredient of the proof is to obtain the
localized estimate of , whose kernel is heavily affected by the physical magnetic diffraction,
and more singular than the classical Bochner-Riesz means
for the Laplacian in .Comment: Comments are welcome! 37 page
The Real-Valued Bochner integral and the Modern Real-Valued Measurable function on
No full textThe like-Lebesgue integral of real-valued measurable functions (abbreviated
as \textit{RVM-MI})is the most complete and appropriate integration Theory.
Integrals are also defined in abstract spaces since Pettis (1938). In
particular, Bochner integrals received much interest with very recent
researches. It is very commode to use the \textit{RVM-MI} in constructing
Bochner integral in Banach or in locally convex spaces. In this simple not, we
prove that the Bochner integral and the \textit{RVM-MI} with respect to a
finite measure are the same on . Applications of that equality
may be useful in weak limits on Banach space.Comment: 11 page
Bochner identities for Fourier transforms
No full textLet G be a compact Lie group and R an orthogonal representation of G acting on
R
n
{{\mathbf {R}}^n}
. For any irreducible unitary representation
π
\pi
of G and vector v in the representation space of
π
\pi
define
S
(
π
,
v
)
\mathcal {S}(\pi ,v)
to be those functions in
S
(
R
n
)
\mathcal {S}({{\mathbf {R}}^n})
which transform (under the action R) according to the vector v. The Fourier transform
F
\mathcal {F}
preserves the class
S
(
π
,
v
)
\mathcal {S}(\pi ,v)
. A Bochner identity asserts that for different choices of G, R,
π
,
v
\pi ,v
the Fourier transform is the same (up to a constant multiple). It is proved here that for G, R,
π
,
v
\pi ,v
and
G
′
,
R
′
,
π
′
,
v
′
G’,R’,\pi ’,v’
and a map
T
:
S
(
π
,
v
)
→
S
(
π
′
,
v
′
)
T:\mathcal {S}(\pi ,v) \to \mathcal {S}(\pi ’,v’)
which has the form: restriction to a subspace followed by multiplication by a fixed function, a Bochner identity
F
′
T
f
=
c
T
F
f
\mathcal {F}’Tf = cT\mathcal {F}f
for all
f
∈
S
(
π
,
v
)
f \in \mathcal {S}(\pi ,v)
holds if and only if
Δ
′
T
f
=
c
1
T
Δ
f
\Delta ’Tf = {c_1}T\Delta f
for all
f
∈
S
(
π
,
v
)
f \in \mathcal {S}(\pi ,v)
. From this result all known Bochner identities follow (due to Harish-Chandra, Herz and Gelbart), as well as some new ones.</p
E-BOCHNER CURVATURE TENSOR ON ALMOST C(λ) MANIFOLDS
No full textThe present paper deals the study of E-Bochner curvature tensor on an almost C(λ) manifolds with the conditions Be (ξ, X).S = 0, Be (ξ, X).R = 0, R.Be (ξ, X) = 0 and Be (ξ, X).Be = 0, where R, S and Be denotes Riemannian curvature tensor, Ricci tensor and E-Bochner curvature tensor, respectively. Also, we have studied ξ-E-Bochner flat C(λ) manifolds
Some estimates for Bochner–Riesz operators on the weighted Herz-type Hardy spaces
No full textAbstractIn this paper, by using the atomic decomposition and molecular characterization of the homogeneous and non-homogeneous weighted Herz-type Hardy spaces HK˙qα,p(w1,w2) (HKqα,p(w1,w2)), we obtain some weighted boundedness properties of the Bochner–Riesz operator and the maximal Bochner–Riesz operator on these spaces for α=n(1/p−1/q), 0<p⩽1 and 1<q<∞
A Note on Endpoint Bochner-Riesz Estimates
No full textWe revisit an -removal argument of Tao to obtain sharp estimates for sums of Bochner-Riesz bumps which are conditional on non-endpoint bounds for single scale bumps. These can be used to obtain sharp conditional sparse bounds for Bochner-Riesz multipliers at the critical index, refining the conditional weak-type estimates of Tao
Weighted estimates for Bochner-Riesz operators on Lorentz spaces
No full textWe present new estimates in the setting of weighted Lorentz spaces of operators satisfying a limited Rubio de Francia condition; namely T is bounded on Lp(v) for every v in an strictly smaller class of weights than the Muckenhoupt class Ap. Important examples will be the Bochner–Riesz operators BRλ with 0<λ<(n−1)/2, sparse operators, Hörmander multipliers with a limited regularity condition and rough operators with Ω∈Lr(Σ) , 1<r<∞.Depto. de Análisis Matemático y Matemática AplicadaFac. de Ciencias MatemáticasTRUEpu
- …
