Browsing by Subject "THEOREM"

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  • Onninen, Jani; Pankka, Pekka (2021)
    We show that, for each 1 S-3 in the Sobolev class W-1,W-p(S-3, S-3).
  • Li, Kangwei; Martikainen, Henri; Vuorinen, Emil (2020)
    We prove some Bloom type estimates in the product BMO setting. More specifically, for a bounded singular integral T-n in R-n and a bounded singular integral T-m in R-m we prove parallel to T-n(1), [b, T-m(2)]]parallel to(Lp(mu)-> Lp(lambda)) less than or similar to([mu]Ap, [lambda]Ap) parallel to b parallel to(BMOprod)(nu), where p is an element of (1, infinity), mu, lambda is an element of A(p) and nu := mu(1/p) lambda(-1/p) is the Bloom. weight. Here T-n(1) is T-n acting on the first variable, T-m(2) is T-m acting on the second variable, A(p) stands for the bi-parameter weights of R-n x R-m and BMOprod(nu) is weighted product BMO space.
  • Li, Kangwei; Martikainen, Henri; Vuorinen, Emil (2021)
    Utilising some recent ideas from our bilinear bi-parameter theory, we give an efficient proof of a two-weight Bloom-type inequality for iterated commutators of linear biparameter singular integrals. We prove that if T is a bi-parameter singular integral satisfying the assumptions of the bi-parameter representation theorem, then parallel to[b(k), center dot center dot center dot [b(2), [b(1), T]] center dot center dot center dot]parallel to(Lp(mu)-> Lp(lambda)) less than or similar to ([mu]Ap,[lambda]Ap) Pi(k)(i=1)parallel to b(i)parallel to(bmo)(nu(theta)i), where p epsilon (1,infinity), theta(i) epsilon [0, 1], Sigma(k)(i=1) theta(i) = 1, mu, lambda epsilon A(p), nu := mu(1/p)lambda(-1/p). Here A(p) stands for the bi-parameter weights in R-n x R-m, and bmo(nu) is a suitable weighted little BMO space. We also simplify the proof of the known 1st order case.
  • Helin, T.; Lassas, M.; Oksanen, L.; Saksala, T. (2018)
    Passive imaging refers to problems where waves generated by unknown sources are recorded and used to image the medium through which they travel. The sources are typically modelled as a random variable and it is assumed that some statistical information is available. In this paper we study the stochastic wave equation partial derivative(2)(t)u- Delta(g)u = chi W, where W is a random variable with the white noise statistics on R1+n, n >= 3, chi is a smooth function vanishing for negative times and outside a compact set in space, and Delta(g) is the Laplace Beltrami operator associated to a smooth non-trapping Riemannian metric tensor g on R-n. The metric tensor g models the medium to be imaged, and we assume that it coincides with the Euclidean metric outside a compact set. We consider the empirical correlations on an open set chi subset of R-n, C-T(t(1), x(1), t(2), x(2)) = 1/T integral(T)(0) u(t(1) s, x(1))u(t(2) s, x(2))ds, t(1), t(2) > 0, x(1), x(2) is an element of chi, for T > 0. Supposing that chi is non-zero on chi and constant in time after t > 1, we show that in the limit T -> infinity, the data C-T becomes statistically stable, that is, independent of the realization of W. Our main result is that, with probability one, this limit determines the Riemannian manifold (R-n, g) up to an isometry. (C) 2018 Elsevier Masson SAS. All rights reserved.
  • Korpela, Jussi; Lassas, Matti; Oksanen, Lauri (2019)
    An inverse boundary value problem for the 1+1 dimensional wave equation (partial derivative(2)(t) - c(x)(2)partial derivative(2)(x))u(x,t) = 0, x is an element of R+ is considered. We give a discrete regularization strategy to recover wave speed c(x) when we are given the boundary value of the wave, u(0,t), that is produced by a single pulse-like source. The regularization strategy gives an approximative wave speed (c) over tilde, satisfying a Holder type estimate parallel to (c) over tilde - c parallel to
  • Guerra, André; Koch, Lukas; Lindberg, Sauli (2021)
    We consider the class of planar maps with Jacobian prescribed to be a fixed radially symmetric function f and which, moreover, fixes the boundary of a ball; we then study maps which minimise the 2p-Dirichlet energy in this class. We find a quantity lambda[f] which controls the symmetry, uniqueness and regularity of minimisers: if lambda[f]
  • Lassas, Matti; Uhlmann, Gunther; Wang, Yiran (2018)
    We consider inverse problems in space-time (M, g), a 4-dimensional Lorentzian manifold. For semilinear wave equations square(g)u + H (x, u) = f, where square(g) denotes the usual Laplace-Beltrami operator, we prove that the source-to-solution map , L : f -> u broken vertical bar v, where V is a neighborhood of a time-like geodesic mu, determines the topological, differentiable structure and the conformal class of the metric of the space-time in the maximal set, where waves can propagate from mu and return back. Moreover, on a given space-time (M, g), the source-to-solution map determines some coefficients of the Taylor expansion of H in u.
  • Greenleaf, Allan; Kurylev, Yaroslav; Lassas, Matti; Uhlmann, Gunther (2008)
  • Di Plinio, Francesco; Li, Kangwei; Martikainen, Henri; Vuorinen, Emil (2020)
    We prove L-p bounds for the extensions of standard multilinear Calderon-Zygmund operators to tuples of UMD spaces tied by a natural product structure. The product can, for instance, mean the pointwise product in UMD function lattices, or the composition of operators in the Schatten-von Neumann subclass of the algebra of bounded operators on a Hilbert space. We do not require additional assumptions beyond UMD on each space-in contrast to previous results, we e.g. show that the Rademacher maximal function property is not necessary. The obtained generality allows for novel applications. For instance, we prove new versions of fractional Leibniz rules via our results concerning the boundedness of multilinear singular integrals in non-commutative L-p spaces. Our proof techniques combine a novel scheme of induction on the multilinearity index with dyadic-probabilistic techniques in the UMD space setting.
  • Hänninen, Timo S.; Verbitsky, Igor E. (2021)
  • Lerner, Andrei K.; Lorist, Emiel; Ombrosi, Sheldy (2022)
    We obtain a sparse domination principle for an arbitrary family of functions f (x, Q), where x is an element of R-n and Q is a cube in R-n. When applied to operators, this result recovers our recent works [37, 39]. On the other hand, our sparse domination principle can be also applied to non-operator objects. In particular, we show applications to generalised Poincare-Sobolev inequalities, tent spaces and general dyadic sums. Moreover, the flexibility of our result allows us to treat operators that are not localisable in the sense of [39], as we will demonstrate in an application to vectorvalued square functions.
  • Fefferman, Charles; Ivanov, Sergei; Kurylev, Yaroslav; Lassas, Matti; Narayanan, Hariharan (2020)
    We study the geometric Whitney problem on how a Riemannian manifold (M, g) can be constructed to approximate a metric space (X, d(X)). This problem is closely related to manifold interpolation (or manifold reconstruction) where a smooth n-dimensional submanifold S subset of R-m, m > n needs to be constructed to approximate a point cloud in Rm. These questions are encountered in differential geometry, machine learning, and in many inverse problems encountered in applications. The determination of a Riemannian manifold includes the construction of its topology, differentiable structure, and metric. We give constructive solutions to the above problems. Moreover, we characterize the metric spaces that can be approximated, by Riemannian manifolds with bounded geometry: We give sufficient conditions to ensure that a metric space can be approximated, in the Gromov-Hausdorff or quasi-isometric sense, by a Riemannian manifold of a fixed dimension and with bounded diameter, sectional curvature, and injectivity radius. Also, we show that similar conditions, with modified values of parameters, are necessary. As an application of the main results, we give a new characterization of Alexandrov spaces with two-sided curvature bounds. Moreover, we characterize the subsets of Euclidean spaces that can be approximated in the Hausdorff metric by submanifolds of a fixed dimension and with bounded principal curvatures and normal injectivity radius. We develop algorithmic procedures that solve the geometric Whitney problem for a metric space and the manifold reconstruction problem in Euclidean space, and estimate the computational complexity of these procedures. The above interpolation problems are also studied for unbounded metric sets and manifolds. The results for Riemannian manifolds are based on a generalization of the Whitney embedding construction where approximative coordinate charts are embedded in R-m and interpolated to a smooth submanifold.
  • Hanninen, Timo S. (2017)
    In this note, we extend Lerner's local median oscillation decomposition to arbitrary (possibly non-doubling) measures. In the light of the analogy between median and mean oscillation, our extension can be viewed as a median oscillation decomposition adapted to the dyadic (martingale) BMO. As an application of the decomposition, we give an alternative proof for the dyadic (martingale) John-Nirenberg inequality, and for Lacey's domination theorem, which states that each martingale transform is pointwise dominated by a positive dyadic operator of zero complexity. Furthermore, by using Lacey's recent technique, we give an alternative proof for Conde-Alonso and Rey's domination theorem, which states that each positive dyadic operator of arbitrary complexity is pointwise dominated by a positive dyadic operator of zero complexity.
  • Hytönen, Tuomas; Li, Kangwei; Sawyer, Eric (2021)
    We prove that for certain positive operators T, such as the Hardy-Littlewood maximal function and fractional integrals, there is a constant D > 1, depending only on the dimension n, such that the two weight norm inequality integral(Rn) T (f sigma)(2) d omega holds for all f >= 0 if and only if the (fractional) A(2) condition holds, and the restricted testing condition integral(Q) T (1 Q(sigma))(2) d omega holds for all cubes Q satisfying vertical bar 2Q|(sigma) = D vertical bar Q vertical bar(sigma). If T is linear, we require as well that the dual restricted testing condition integral(Q) T* (1(Q)holds for all cubes Q satisfying vertical bar 2Q vertical bar(omega)
  • Peltola, T.; Eremin, V.; Verbitskaya, E.; Härkönen, J. (2017)
    Segmented silicon detectors (micropixel and microstrip) are the main type of detectors used in the inner trackers of Large Hadron Collider (LHC) experiments at CERN. Due to the high luminosity and eventual high fluence of energetic particles, detectors with fast response to fit the short shaping time of 20-25 ns and sufficient radiation hardness are required. Charge collection measurements carried out at the Ioffe Institute have shown a reversal of the pulse polarity in the detector response to short-range charge injection. Since the measured negative signal is about 30-60% of the peak positive signal, the effect strongly reduces the CCE even in non-irradiated detectors. For further investigation of the phenomenon the measurements have been reproduced by TCAD simulations. As for the measurements, the simulation study was applied for the p-on-n strip detectors similar in geometry to those developed for the ATLAS experiment and for the Ioffe Institute designed p-on-n strip detectors with each strip having a window in the metallization covering the p(+) implant, allowing the generation of electron-hole pairs under the strip implant. Red laser scans across the strips and the interstrip gap with varying laser diameters and Si-SiO2 interface charge densities (Q(f)) were carried out. The results verify the experimentally observed negative response along the scan in the interstrip gap. When the laser spot is positioned on the strip p(+) implant the negative response vanishes and the collected charge at the active strip increases respectively. The simulation results offer a further insight and understanding of the influence of the oxide charge density in the signal formation. The main result of the study is that a threshold value of Q(f), that enables negligible losses of collected charges, is defined. The observed effects and details of the detector response for different charge injection positions are discussed in the context of Ramo's theorem.
  • Hytonen, Tuomas P. (2021)
    Supplying the missing necessary conditions, we complete the characterisation of the L-p -> L-q boundedness of commutators [b, T] of pointwise multiplication and Calderon-Zygmund operators, for arbitrary pairs of 1 < p, q For p For p > q, our results are new even for special classical operators with smooth kernels. As an application, we show that every f is an element of L-p(R-d) can be represented as a convergent series of normalised Jacobians J(u) = det del uof u is an element of (over dot(W))(1,dp)(R-d)(d). This extends, from p = 1 to p > 1, a result of Coifman, Lions, Meyer and Semmes about J:. (over dot(W))(1,d)(R-d)(d) -> H-1(R-d), and supports a conjecture of Iwaniec about the solvability of the equation Ju = f is an element of L-p(R-d). (C) 2021 The Author(s). Published by Elsevier Masson SAS.
  • Hänninen, Timo S. (2017)
    We study the operator-valued positive dyadic operator T-lambda (f sigma) := Sigma(Q is an element of D) lambda(Q) integral(Q) f d sigma 1(Q,) where the coefficients {lambda(Q) : C -> D}(Q is an element of D) are positive operators from a Banach lattice C to a Banach lattice D. We assume that the Banach lattices C and D* each have the Hardy-Littlewood property. An example of a Banach lattice with the Hardy-Littlewood property is a Lebesgue space. In the two-weight case, we prove that the L-C(p) (sigma) -> L-D(q)(omega) boundedness of the operator T-lambda(. sigma) is characterized by the direct and the dual L-infinity testing conditions: parallel to 1(Q)T(lambda) (1(Q)f sigma)parallel to(LDq) ((omega)) less than or similar to parallel to f parallel to(LC infinity) ((Q,sigma)) sigma(Q)(1/p), parallel to 1(Q)T(lambda)* (1(Q)g omega)parallel to(LC*p') ((sigma)) less than or similar to parallel to g parallel to(LD*infinity) ((Q,omega)) omega(Q)(1/q'), Here L-C(p) (sigma) and L-D(q) (omega) denote the Lebesgue-Bochner spaces associated with exponents 1 <p In the unweighted case, we show that the L-C(p) (mu) -> L-D(p) (mu) boundedness of the operator T-lambda(. mu) is equivalent to the end-point direct L-infinity testing condition: parallel to 1(Q)T(lambda) (1(Q)f mu)parallel to(LD1) (mu) less than or similar to parallel to f parallel to(LC infinity) ((Q,mu)) mu(Q). This condition is manifestly independent of the exponent p. By specializing this to particular cases, we recover some earlier results in a unified way.