Interpolation of Operators - "the real method"

• Week 2
    Wed (9/01): The functional f^**; Strong-type operator interpolation for (L¹,L^∞); the Peetre K-functional; connection with f^** and the real method of interpolation of operators.
    Fri (9/03): The Lebesgue L^p and Lorentz spaces L^p,q spaces. Completeness of the general intermediate Banach spaces (X₁,X₂)_θ,q.

Hardy-Littlewood Maximal Operator and Approximate Identities

• Week 3
    Mon (9/06): Labor Day Holiday.
    Wed (9/08): Hardy integral inequalities for the dual averaging operators P and Q; Defn. of the Hardy-Littlewood Maximal Operator M[f]; an elementary covering lemma for measurable sets in R^d.
• Week 4
    Mon (9/13): Weak type (1,1) estimate; bounded properties of the Maximal Function on Lebesgue and Lorentz spaces. Variations of the maximal operator. Control of Almost Everywhere Convergence with respect to averages over general measurable sets. Lebesgue points. Convolution of integrable functions.
    Wed: Classes Postponed due to Hurricane
    Fri (9/17): Approximations to the Identity - Control by Hardy-Littlewood maximal operator (for those with integrable radial majorants), almost everywhere convergence. Examples of Poisson and Gauss-Weierstrass kernels.

Calderòn's maximal operator and weak-type operators

• Week 5
    Mon (9/20): The L^p-modulus of continuity, Convergence in norm of approximations to the identity; Solution of Dirichlet boundary value problems for the upper half space R^d₊.
    Wed (9/22): Talk by Exxon-Mobil.
    Fri (9/24): General weak type interpolation theory for spaces of measurable functions; Calderòn's operator S_σ[f] ; Maximal property of Calderòn operator.

Fourier Transform (L¹ and L² theory); Special Techniques.

• Week 6
    Mon (9/27): Embeddings of Lorentz spaces for R, T^d, Z, and R^d. Mapping properties of Calderòn Operator; Young's inequality for convolutions; Introduction to Fourier transform for R^d; basic properties;
    Wed (9/29): Fourier Transforms of some important functions - e.g. Gaussian as an eigenfunction; the Fourier inversion theorem for L²; Plancherel Theorem; the Hausdorff-Young inequality; Paley's extension.

Some Applications to PDE and Fractional Calculus.

• Week 7
    Mon (10/04): Fourier multipliers; Solution to heat equation and Dirichlet's problem for upper half spaces. Semigroup properties.
    Wed (10/06): Fractional derivatives/integrals and the Hardy-Littlewood-Sobolev inequality. The conjugate Poisson kernel, its associated multiplier, and the motivation for singular integral operators.

Singular Integral Operators - I

• Week 8
    Mon (10/11): Fall Break
    Wed (10/13): Motivation for Singular Integral Operators - sketch of the relationship between the Poisson kernel, the conjugate Poisson kernel and the Hilbert transform; "stopping time" covering and the Calderòn-Zygmund decomposition of an integrable function into a sum of "good" and "bad" functions.

Singular Integral Operators - II

• Week 9
    Mon (10/18): The Whitney covering lemma for open sets; the corresponding Calderòn-Zygmund decomposition and its properties.
    Wed (10/20): Class rescheduled for Friday.
    Fri (10/22): A first theorem on singular integral operators: Strong type (2,2) and weak type (1,1) inequalities; the Marcinkiewicz integral and its use in the weak type (1,1) estimate. General L^p results.

Singular Integral Operators - III

• Week 10
    Mon (10/25): Application to the conjugate Poisson kernel; Variants of hypotheses for the boundedness of the singular integral operators on L^p(R^d):
  • kernels K satisfying the Hörmander, cancellation, and growth conditions.
  • dilation-invariant singular integral operators, i.e. kernels of the form W(x)/ |x|^d homogeneous of degree -d: Dini type smoothness condition and cancellation.
  
    Wed (10/27): Calculation of multiplier symbols for kernels homogeneous of degree -d; the Riesz transforms R_j(f) (j=1,...,d) and their multipliers; an application on mixed derivatives.

Littlewood-Paley g-functions I

• Week 11
    Mon (11/01): Introduction to Littlewood-Paley g-functions and the L² theory; Banach space-valued functions, strong and weak measurability; the Bochner integral.
    Wed (11/03): Hilbert space-valued measurable functions and Young's convolution inequality in this context; Hilbert space-valued singular integral operators with operator-valued kernels.

Littlewood-Paley g-functions II

• Week 12
    Mon (11/08): Littlewood-Paley g-functions realized as the H₂-norm of the singular integral operator with kernel K(·,y)= ∇^˜P_y(·) applied to the function f. Verification of conditions to guarantee L^p(R^d,H₂)- boundness of the operator. Lower estimates through duality and polarization of the L² identities.
    Wed (11/10): Generalized Cauchy-Riemann conditions and conjugate harmonic systems. Variants of Littlewood-Paley functions: g^(k)[f](x), the square function, and g_λ^*[f](x) and their relationships.

Littlewood-Paley g-functions III

• Week 13
    Mon (11/15): Proofs of relationships among the Littlewood-Paley g-functions, boundedness of the g_λ^* function on L^p(R^d).
    Wed (11/17): Lecture 24 (in PowerPoint) Current Research Applications, the Fourier and Haar bases and some simple properties. Burn-In and a Tree Encoder.

Introduction to Besov Spaces (univariate)

• Week 14
    Mon (11/22): Review classical definition of the modulus of smoothness; Additional properties of moduli; Marchaud's inequality.
    Wed (11/17): University Holiday - Thanksgiving

• Week 15
    Mon (11/29): Characterization of the modulus of smoothness in L^p as the Peetre K-functional between L^p and the Sobolev space W^r(L^p); B-splines; the intermediate theorem for derivatives; Besov spaces as interpolation spaces between L^p and W^r(L^p).
    Wed (12/01): Sobolev Embedding Theorems for Sobolev and Besov spaces; weak type inequalities relating decreasing rearrangements and moduli of continuity. Preview of Math 758S.