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  • 1.
    Aleksanyan, Hayk
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.). The University of Edinburgh.
    Shahgholian, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Sjölin, Per
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Applications of Fourier analysis in homogenization of Dirichlet problem I. Pointwise estimates2013In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 254, no 6, p. 2626-2637Article in journal (Refereed)
    Abstract [en]

    In this paper we prove convergence results for homogenization problem for solutions of partial differential system with rapidly oscillating Dirichlet data. Our method is based on analysis of oscillatory integrals. In the uniformly convex and smooth domain, and smooth operator and boundary data, we prove pointwise convergence results, namely vertical bar u(epsilon)(x) - u(0)(x)vertical bar <= C-kappa epsilon((d-1)/2) 1/d(x)(kappa), for all x is an element of D, for all kappa > d - 1, where u(epsilon) and u(0) are solutions of respectively oscillating and homogenized Dirichlet problems, and d(x) is the distance of x from the boundary of D. As a corollary for all 1 <= p < infinity we obtain L-P convergence rate as well.

  • 2.
    Aleksanyan, Hayk
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.). The University of Edinburgh.
    Shahgholian, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Sjölin, Per
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Applications of Fourier Analysis in Homogenization of Dirichlet Problem III: Polygonal Domains2014In: Journal of Fourier Analysis and Applications, ISSN 1069-5869, E-ISSN 1531-5851, Vol. 20, no 3, p. 524-546Article in journal (Refereed)
    Abstract [en]

    In this paper we prove convergence results for the homogenization of the Dirichlet problem for elliptic equations in divergence form with rapidly oscillating boundary data and non oscillating coefficients in convex polygonal domains. Our analysis is based on integral representation of solutions. Under a certain Diophantine condition on the boundary of the domain and smooth coefficients we prove pointwise, as well as convergence results. For larger exponents we prove that the convergence rate is close to optimal. We also suggest several directions of possible generalization of the results in this paper.

  • 3.
    Aleksanyan, Hayk
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.). The University of Edinburgh.
    Shahgholian, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Sjölin, Per
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Applications of Fourier Analysis in Homogenization of the Dirichlet Problem: L-p Estimates2015In: Archive for Rational Mechanics and Analysis, ISSN 0003-9527, E-ISSN 1432-0673, Vol. 215, no 1, p. 65-87Article in journal (Refereed)
    Abstract [en]

    Let u(epsilon) be a solution to the system div(A(epsilon)(x)del u(epsilon)(x)) = 0 in D, u(epsilon)(x) = g(x, x/epsilon) on partial derivative D, where D subset of R-d (d >= 2), is a smooth uniformly convex domain, and g is 1-periodic in its second variable, and both A(epsilon) and g are sufficiently smooth. Our results in this paper are twofold. First we prove L-p convergence results for solutions of the above system and for the non-oscillating operator A(epsilon)(x) = A(x), with the following convergence rate for all 1 <= p < infinity parallel to u(epsilon) - u(0)parallel to (LP(D)) <= C-P {epsilon(1/2p), d = 2, (epsilon vertical bar ln epsilon vertical bar)(1/p), d = 3, epsilon(1/p), d >= 4, which we prove is (generically) sharp for d >= 4. Here u(0) is the solution to the averaging problem. Second, combining our method with the recent results due to Kenig, Lin and Shen (Commun Pure Appl Math 67(8): 1219-1262, 2014), we prove (for certain class of operators and when d >= 3) ||u(epsilon) - u(0)||(Lp(D)) <= C-p[epsilon(ln(1/epsilon))(2)](1/p) for both the oscillating operator and boundary data. For this case, we take A(epsilon) = A(x/epsilon), where A is 1-periodic as well. Some further applications of the method to the homogenization of the Neumann problem with oscillating boundary data are also considered.

  • 4.
    Aleksanyan, Hayk
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.). The University of Edinburgh.
    Shahgholian, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Sjölin, Per
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    L2-estimates for singular oscillatory integral operators2016In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 441, no 2, p. 529-548Article in journal (Refereed)
    Abstract [en]

    In this note we study singular oscillatory integrals with linear phase function over hypersurfaces which may oscillate, and prove estimates of L2L2 type for the operator, as well as for the corresponding maximal function. If the hypersurface is flat, we consider a particular class of a nonlinear phase functions, and apply our analysis to the eigenvalue problem associated with the Helmholtz equation in R3.

  • 5.
    Sjölin, Per
    KTH, Superseded Departments, Mathematics.
    A Littlewood-Paley inequality for the Carleson operator2000In: Journal of Fourier Analysis and Applications, ISSN 1069-5869, E-ISSN 1531-5851, Vol. 6, no 5, p. 457-466Article in journal (Refereed)
    Abstract [en]

    The Carleson operator is closely related to the maximal partial sum operator for Fourier series. We study generalizations of this operator in one and several variables.

  • 6.
    Sjölin, Per
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Convergence of generalized conjugate partial Fourier integrals2007In: Mathematische Zeitschrift, ISSN 0025-5874, E-ISSN 1432-1823, Vol. 256, no 2, p. 265-278Article in journal (Refereed)
    Abstract [en]

    Generalized conjugate partial Fourier integrals are used to find jumps of functions. The rate of convergence is studied and sharp results are obtained.

  • 7.
    Sjölin, Per
    KTH, Superseded Departments, Mathematics.
    Estimates of averages of Fourier transforms with respect to general measures2003In: Proceedings of the Royal Society of Edinburgh. Section A Mathematics, ISSN 0308-2105, E-ISSN 1473-7124, Vol. 133, p. 943-950Article in journal (Refereed)
    Abstract [en]

    We study a connection between the L-2 average decay of the Fourier transform of functions with respect to a given measure and the Hausdorff behaviour of that measure.

  • 8.
    Sjölin, Per
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Heisenberg uniqueness pairs and a theorem of Beurling and Malliavin2011In: Bulletin des Sciences Mathématiques, ISSN 0007-4497, E-ISSN 1952-4773, Vol. 135, no 2, p. 125-133Article in journal (Refereed)
    Abstract [en]

    Let T denote the unit circle in the plane. For various simple sets Lambda in the plane we shall study the question whether (T, Lambda) is a Heisenberg uniqueness pair. For example, we shall consider the cases where Lambda is a circle or a union of two straight lines. We shall also use a theorem of Beurling and Malliavin.

  • 9.
    Sjölin, Per
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Heisenberg Uniqueness Pairs for the Parabola2013In: Journal of Fourier Analysis and Applications, ISSN 1069-5869, E-ISSN 1531-5851, Vol. 19, no 2, p. 410-416Article in journal (Refereed)
    Abstract [en]

    Let I" denote the parabola y=x (2) in the plane. For some simple sets I > in the plane we study the question whether (I",I >) is a Heisenberg uniqueness pair. For example we shall consider the cases where I > is a straight line or a union of two straight lines.

  • 10.
    Sjölin, Per
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    L-p-Estimates for Singular Oscillatory Integral Operators2017In: Journal of Fourier Analysis and Applications, ISSN 1069-5869, E-ISSN 1531-5851, Vol. 23, no 6, p. 1408-1425Article in journal (Refereed)
    Abstract [en]

    In this paper we study singular oscillatory integrals with a nonlinear phase function. We prove estimates of and type.

  • 11.
    Sjölin, Per
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Maximal estimates for solutions to the nonelliptic Schrödinger equation2007In: Bulletin of the London Mathematical Society, ISSN 0024-6093, E-ISSN 1469-2120, Vol. 39, p. 404-412Article in journal (Refereed)
    Abstract [en]

    Maximal estimates are studied for solutions to an initial value problem for the nonelliptic Schrodinger equation. A result of Rogers, Vargas and Vega is extended.

  • 12.
    Sjölin, Per
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Maximal operators of Schrödinger type with a complex parameter2009In: Mathematica Scandinavica, ISSN 0025-5521, E-ISSN 1903-1807, Vol. 105, no 1, p. 121-133Article in journal (Refereed)
    Abstract [en]

    Maximal operators of Schrödinger type but with a complex parameter are considered. For these operators we obtain results which in a certain sense lie between the results for the corresponding maximal operators for solutions to the Schrödinger equation and for solutions to the heat equation.

  • 13.
    Sjölin, Per
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Nonlocalization of operators of schrödinger type2013In: Annales Academiae Scientiarum Fennicae Mathematica, ISSN 1239-629X, E-ISSN 1798-2383, Vol. 38, no 1, p. 141-147Article in journal (Refereed)
    Abstract [en]

    Localization properties are studied for operators of Schrodinger type.

  • 14.
    Sjölin, Per
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Radial Functions and Maximal Operators of Schrodinger Type2011In: Indiana University Mathematics Journal, ISSN 0022-2518, E-ISSN 1943-5258, Vol. 60, no 1, p. 143-159Article in journal (Refereed)
    Abstract [en]

    Estimates are considered for maximal operators of Schrodinger type acting on radial functions. Sharp and almost sharp results are obtained.

  • 15.
    Sjölin, Per
    KTH, Superseded Departments, Mathematics.
    Remarks on a theorem by N. Yu. Antonov2003In: Studia Mathematica, ISSN 0039-3223, E-ISSN 1730-6337, Vol. 158, no 1, p. 79-97Article in journal (Refereed)
    Abstract [en]

    We extend some results of N. Yu. Antonov on convergence of Fourier series to more general settings. One special feature of our work is that we do not assume smoothness for the kernels in our hypotheses. This has interesting applications to convergence with respect to general orthonormal systems, like the Walsh-Fourier system, for which we prove a.e. convergence in the class L log L log log log L. Other applications are given in the theory of differentiation of integrals.

  • 16.
    Sjölin, Per
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Some remarks on localization of Schrodinger means2012In: Bulletin des Sciences Mathématiques, ISSN 0007-4497, E-ISSN 1952-4773, Vol. 136, no 6, p. 638-647Article in journal (Refereed)
    Abstract [en]

    We study localization and localization almost everywhere of Schrodinger means of functions in Sobolev spaces.

  • 17.
    Sjölin, Per
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Some remarks on singular oscillatory integrals and convolution operators2017In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 145, no 9, p. 3843-3848Article in journal (Refereed)
    Abstract [en]

    In this note we study the relation between oscillatory integral operators and convolution operators, and also the sharpness of L-p-estimates for singular oscillatory integral operators.

  • 18.
    Sjölin, Per
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Some remarks on Sobolev regularity2009In: Acta Scientarum Mathematicarum, ISSN 0001-6969, Vol. 75, no 1-2, p. 233-239Article in journal (Refereed)
    Abstract [en]

    Some characterizations of Sobolev spaces are discussed.

  • 19.
    Sjölin, Per
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Spherical harmonics and maximal estimates for the Schrodinger equation2005In: Annales Academiae Scientiarum Fennicae Mathematica, ISSN 1239-629X, E-ISSN 1798-2383, Vol. 30, no 2, p. 393-406Article in journal (Refereed)
    Abstract [en]

    Maximal estimates are considered for solutions to an initial value problem for the Schrodinger equation. The initial value function is assumed to be a linear combination of products of radial functions and spherical harmonics. This generalizes the case of radial functions. We also replace the solutions to the Schrodinger equation by more general oscillatory integrals.

  • 20.
    Sjölin, Per
    KTH, Superseded Departments, Mathematics.
    Spherical harmonics and spherical averages of Fourier transforms2002In: Rendiconti del Seminario Matematico della Universita di Padova, ISSN 0041-8994, E-ISSN 2240-2926, Vol. 108, p. 41-51Article in journal (Refereed)
    Abstract [en]

    We give estimates for spherical averages of Fourier transforms of functions which are linear combinations of products of radial functions and spherical harmonics. This generalizes the case of radial functions.

  • 21.
    Sjölin, Per
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Soria, Fernando
    A note on multiparameter maximal operators of Schrödinger type2010In: Revista Matemática Complutense, ISSN 1139-1138, Vol. 23, no 1, p. 261-265Article in journal (Refereed)
    Abstract [en]

    In recent years, new features about the boundedness properties of maximal operators related to the solutions of the Schrodinger equation have evolved by considering different time values for each variable. In this paper, a multiparameter maximal estimate is studied for operators of Schrodinger type. Sharp results are obtained.

  • 22.
    Sjölin, Per
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Soria, Fernando
    A note on Schrödinger maximal operators with a complex parameter2010In: Journal of the Australian Mathematical Society, ISSN 1446-7887, E-ISSN 1446-8107, Vol. 88, no 3, p. 405-412Article in journal (Refereed)
    Abstract [en]

    Extending previous results of the first author, some new estimates are obtained for maximal operators of Schrodinger type with a complex parameter.

  • 23.
    Sjölin, Per
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Soria, Fernando
    Estimates for multiparameter maximal operators of Schrödinger type2014In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 411, no 1, p. 129-143Article in journal (Refereed)
    Abstract [en]

    Multiparameter maximal estimates are considered for operators of Schrodinger type. Sharp and almost sharp results, that extend work by Rogers and Villarroya, are obtained. We provide new estimates via the integrability of the kernel which naturally appears with a TT* argument and discuss the behavior at the endpoints. We treat in particular the case of global integrability of the maximal operator on finite time for solutions to the linear Schrodinger equation and make some comments on an open problem.

1 - 23 of 23
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