Displaying similar documents to “Estimates with global range for oscillatory integrals with concave phase”

L p type mapping estimates for oscillatory integrals in higher dimensions

G. Sampson (2006)

Studia Mathematica

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We show in two dimensions that if K f = ² k ( x , y ) f ( y ) d y , k ( x , y ) = ( e i x a · y b ) / ( | x - y | η ) , p = 4/(2+η), a ≥ b ≥ 1̅ = (1,1), v p ( y ) = y ( p / p ' ) ( 1 ̅ - b / a ) , then | | K f | | p C | | f | | p , v p if η + α₁ + α₂ < 2, α j = 1 - b j / a j , j = 1,2. Our methods apply in all dimensions and also for more general kernels.

Nonrectifiable oscillatory solutions of second order linear differential equations

Takanao Kanemitsu, Satoshi Tanaka (2017)

Archivum Mathematicum

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The second order linear differential equation ( p ( x ) y ' ) ' + q ( x ) y = 0 , x ( 0 , x 0 ] is considered, where p , q C 1 ( 0 , x 0 ] , p ( x ) > 0 , q ( x ) > 0 for x ( 0 , x 0 ] . Sufficient conditions are established for every nontrivial solutions to be nonrectifiable oscillatory near x = 0 without the Hartman–Wintner condition.

A radial estimate for the maximal operator associated with the free Schrödinger equation

Sichun Wang (2006)

Studia Mathematica

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Let d > 0 be a positive real number and n ≥ 1 a positive integer and define the operator S d and its associated global maximal operator S * * d by ( S d f ) ( x , t ) = 1 / ( 2 π ) e i x · ξ e i t | ξ | d f ̂ ( ξ ) d ξ , f ∈ (ℝⁿ), x ∈ ℝⁿ, t ∈ ℝ, ( S * * d f ) ( x ) = s u p t | 1 / ( 2 π ) e i x · ξ e i t | ξ | d f ̂ ( ξ ) d ξ | , f ∈ (ℝⁿ), x ∈ ℝⁿ, where f̂ is the Fourier transform of f and (ℝⁿ) is the Schwartz class of rapidly decreasing functions. If d = 2, S d f is the solution to the initial value problem for the free Schrödinger equation (cf. (1.3) in this paper). We prove that for radial functions f ∈ (ℝⁿ), if n ≥ 3, 0 < d ≤ 2, and p ≥...

Global regularity for the 3D MHD system with damping

Zujin Zhang, Xian Yang (2016)

Colloquium Mathematicae

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We study the Cauchy problem for the 3D MHD system with damping terms ε | u | α - 1 u and δ | b | β - 1 b (ε, δ > 0 and α, β ≥ 1), and show that the strong solution exists globally for any α, β > 3. This improves the previous results significantly.

On Kakeya–Nikodym averages, L p -norms and lower bounds for nodal sets of eigenfunctions in higher dimensions

Matthew D. Blair, Christopher D. Sogge (2015)

Journal of the European Mathematical Society

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We extend a result of the second author [27, Theorem 1.1] to dimensions d 3 which relates the size of L p -norms of eigenfunctions for 2 < p < 2 ( d + 1 ) / d - 1 to the amount of L 2 -mass in shrinking tubes about unit-length geodesics. The proof uses bilinear oscillatory integral estimates of Lee [22] and a variable coefficient variant of an " ϵ removal lemma" of Tao and Vargas [35]. We also use Hörmander’s [20] L 2 oscillatory integral theorem and the Cartan–Hadamard theorem to show that, under the assumption of nonpositive...

An improved regularity criteria for the MHD system based on two components of the solution

Zujin Zhang, Yali Zhang (2021)

Applications of Mathematics

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As observed by Yamazaki, the third component b 3 of the magnetic field can be estimated by the corresponding component u 3 of the velocity field in L λ ( 2 λ 6 ) norm. This leads him to establish regularity criterion involving u 3 , j 3 or u 3 , ω 3 . Noticing that λ can be greater than 6 in this paper, we can improve previous results.

Integral averaging technique for oscillation of damped half-linear oscillators

Yukihide Enaka, Masakazu Onitsuka (2018)

Czechoslovak Mathematical Journal

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This paper is concerned with the oscillatory behavior of the damped half-linear oscillator ( a ( t ) φ p ( x ' ) ) ' + b ( t ) φ p ( x ' ) + c ( t ) φ p ( x ) = 0 , where φ p ( x ) = | x | p - 1 sgn x for x and p > 1 . A sufficient condition is established for oscillation of all nontrivial solutions of the damped half-linear oscillator under the integral averaging conditions. The main result can be given by using a generalized Young’s inequality and the Riccati type technique. Some examples are included to illustrate the result. Especially, an example which asserts that all nontrivial...

Fourier approximation and embeddings of Sobolev spaces

D. E. Edmunds, V. B. Moscatelli

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CONTENTSIntroduction............................................................................................................ 51. Preliminaries............................................................................................................. 82. Embedding into W m , p ( Ω ) into L S ( Ω ) (n>1).......................................... 103. The case n = 1.......................................................................................................... 284. Embedding W m , p ( Ω ) into L φ ( Ω ) ...............................................................

Sobolev type inequalities for fractional maximal functions and Riesz potentials in Morrey spaces of variable exponent on half spaces

Yoshihiro Mizuta, Tetsu Shimomura (2023)

Czechoslovak Mathematical Journal

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Our aim is to establish Sobolev type inequalities for fractional maximal functions M , ν f and Riesz potentials I , α f in weighted Morrey spaces of variable exponent on the half space . We also obtain Sobolev type inequalities for a C 1 function on . As an application, we obtain Sobolev type inequality for double phase functionals with variable exponents Φ ( x , t ) = t p ( x ) + ( b ( x ) t ) q ( x ) , where p ( · ) and q ( · ) satisfy log-Hölder conditions, p ( x ) < q ( x ) for x , and b ( · ) is nonnegative and Hölder continuous of order θ ( 0 , 1 ] .

Remarks on the Bourgain-Brezis-Mironescu Approach to Sobolev Spaces

B. Bojarski (2011)

Bulletin of the Polish Academy of Sciences. Mathematics

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For a function f L l o c p ( ) the notion of p-mean variation of order 1, p ( f , ) is defined. It generalizes the concept of F. Riesz variation of functions on the real line ℝ¹ to ℝⁿ, n > 1. The characterisation of the Sobolev space W 1 , p ( ) in terms of p ( f , ) is directly related to the characterisation of W 1 , p ( ) by Lipschitz type pointwise inequalities of Bojarski, Hajłasz and Strzelecki and to the Bourgain-Brezis-Mironescu approach.

A complete characterization of R-sets in the theory of differentiation of integrals

G. A. Karagulyan (2007)

Studia Mathematica

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Let s be the family of open rectangles in the plane ℝ² with a side of angle s to the x-axis. We say that a set S of directions is an R-set if there exists a function f ∈ L¹(ℝ²) such that the basis s differentiates the integral of f if s ∉ S, and D ̅ s f ( x ) = l i m s u p d i a m ( R ) 0 , x R s | R | - 1 R f = almost everywhere if s ∈ S. If the condition D ̅ s f ( x ) = holds on a set of positive measure (instead of a.e.) we say that S is a WR-set. It is proved that S is an R-set (resp. a WR-set) if and only if it is a G δ (resp. a G δ σ ).

Some Hölder-logarithmic estimates on Hardy-Sobolev spaces

Imed Feki, Ameni Massoudi (2024)

Czechoslovak Mathematical Journal

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We prove some optimal estimates of Hölder-logarithmic type in the Hardy-Sobolev spaces H k , p ( G ) , where k * , 1 p and G is either the open unit disk 𝔻 or the annular domain G s , 0 < s < 1 of the complex space . More precisely, we study the behavior on the interior of G of any function f belonging to the unit ball of the Hardy-Sobolev spaces H k , p ( G ) from its behavior on any open connected subset I of the boundary G of G with respect to the L 1 -norm. Our results can be viewed as an improvement and generalization of...

Maximal non-pseudovaluation subrings of an integral domain

Rahul Kumar (2024)

Czechoslovak Mathematical Journal

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The notion of maximal non-pseudovaluation subring of an integral domain is introduced and studied. Let R S be an extension of domains. Then R is called a maximal non-pseudovaluation subring of S if R is not a pseudovaluation subring of S , and for any ring T such that R T S , T is a pseudovaluation subring of S . We show that if S is not local, then there no such T exists between R and S . We also characterize maximal non-pseudovaluation subrings of a local integral domain.

Functions with prescribed singularities

Giovanni Alberti, S. Baldo, G. Orlandi (2003)

Journal of the European Mathematical Society

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The distributional k -dimensional Jacobian of a map u in the Sobolev space W 1 , k 1 which takes values in the sphere S k 1 can be viewed as the boundary of a rectifiable current of codimension k carried by (part of) the singularity of u which is topologically relevant. The main purpose of this paper is to investigate the range of the Jacobian operator; in particular, we show that any boundary M of codimension k can be realized as Jacobian of a Sobolev map valued in S k 1 . In case M is polyhedral, the...

A note on the existence of solutions with prescribed asymptotic behavior for half-linear ordinary differential equations

Manabu Naito (2024)

Mathematica Bohemica

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The half-linear differential equation ( | u ' | α sgn u ' ) ' = α ( λ α + 1 + b ( t ) ) | u | α sgn u , t t 0 , is considered, where α and λ are positive constants and b ( t ) is a real-valued continuous function on [ t 0 , ) . It is proved that, under a mild integral smallness condition of b ( t ) which is weaker than the absolutely integrable condition of b ( t ) , the above equation has a nonoscillatory solution u 0 ( t ) such that u 0 ( t ) e - λ t and u 0 ' ( t ) - λ e - λ t ( t ), and a nonoscillatory solution u 1 ( t ) such that u 1 ( t ) e λ t and u 1 ' ( t ) λ e λ t ( t ).

A regularity theory for scalar local minimizers of splitting-type variational integrals

Michael Bildhauer, Martin Fuchs, Xiao Zhong (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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Starting from Giaquinta’s counterexample [12] we introduce the class of splitting functionals being of ( p , q ) -growth with exponents p q &lt; and show for the scalar case that locally bounded local minimizers are of class C 1 , μ . Note that to our knowledge the only C 1 , μ -results without imposing a relation between p and q concern the case of two independent variables as it is outlined in Marcellini’s paper [15], Theorem A, and later on in the work of Fusco and Sbordone [10], Theorem 4.2.

Capacitary estimates of positive solutions of semilinear elliptic equations with absorbtion

Moshe Marcus, Laurent Véron (2004)

Journal of the European Mathematical Society

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Let Ω be a bounded domain of class C 2 in N and let K be a compact subset of Ω . Assume that q ( N + 1 ) / ( N 1 ) and denote by U K the maximal solution of Δ u + u q = 0 in Ω which vanishes on Ω K . We obtain sharp upper and lower estimates for U K in terms of the Bessel capacity C 2 / q , q ' and prove that U K is σ -moderate. In addition we describe the precise asymptotic behavior of U K at points σ K , which depends on the “density” of K at σ , measured in terms of the capacity C 2 / q , q ' .

Further characterizations of Sobolev spaces

Hoai-Minh Nguyen (2008)

Journal of the European Mathematical Society

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Let ( F n ) n be a sequence of non-decreasing functions from [ 0 , + ) into [ 0 , + ) . Under some suitable hypotheses of ( F n ) n , we will prove that if g L p ( N ) , 1 < p < + , satisfies lim inf n N N F n ( | g ( x ) - g ( y ) | ) / | x - y | N + p d x d y < + , then g W 1 , p ( N ) and moreover lim n N N F n ( | g ( x ) - g ( y ) | ) / | x - y | N + p d x d y = K N , p N | g ( x ) | p d x , where K N , p is a positive constant depending only on N and p . This extends some results in J. Bourgain and H-M. Nguyen [A new characterization of Sobolev spaces, C. R. Acad Sci. Paris, Ser. 343 (2006) 75-80] and H-M. Nguyen [Some new characterizations of Sobolev spaces, J. Funct. Anal. 237 (2006) 689-720]. We also present some...

A priori bounds for positive radial solutions of quasilinear equations of Lane–Emden type

Soohyun Bae (2023)

Archivum Mathematicum

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We consider the quasilinear equation Δ p u + K ( | x | ) u q = 0 , and present the proof of the local existence of positive radial solutions near 0 under suitable conditions on K . Moreover, we provide a priori estimates of positive radial solutions near when r - K ( r ) for - p is bounded near .

Sums of commuting operators with maximal regularity

Christian Le Merdy, Arnaud Simard (2001)

Studia Mathematica

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Let Y be a Banach space and let S L p be a subspace of an L p space, for some p ∈ (1,∞). We consider two operators B and C acting on S and Y respectively and satisfying the so-called maximal regularity property. Let ℬ and be their natural extensions to S ( Y ) L p ( Y ) . We investigate conditions that imply that ℬ + is closed and has the maximal regularity property. Extending theorems of Lamberton and Weis, we show in particular that this holds if Y is a UMD Banach lattice and e - t B is a positive contraction...