We consider a convolution operator in R d with kernel in L q acting from L p to L s , where 1/p + 1/q = 1 + 1/s. The main theorem states that if 1 < q, p, s < ∞, then there exists an L p function of unit norm on which the s-norm of the convolution is attained. A number of questions, related to the statement and proof of the main theorem, are discussed. Also the problem of computing best constants in the Hausdorff-Young inequality for the Laplace transform, which prompted this research, is considered.1 Emphasizing that K p,r is the extremum of the symmetric bilinear form K p,r = K r,p = sup k(x + y)f (x)g(y) dx dy ; f p = g r = 1. 2 The labels (a) , (b) etc. refer to the comments (a), (b) etc. in Section 8. 1. The class Max comprises all maximizing sequences (for the convolution operator K : L p → L r ). 2. The class SMax comprises all special maximizing sequences, that is, maximizing sequences of the form (Bf n ), where (f n ) ∈ Max. 3. The class RTgt comprises all relatively tight sequences. 4. The class Tgt comprises all tight sequences. 5. The class WCvg comprises all weakly convergent sequences. 6. The class LCvg comprises all locally convergent sequences, i.e. sequences converging in L p norm on any bounded measurable subset of R d .

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Let $C=\inf (k/n)\sum_{i=1}^n x_i(x_{i+1}+\dots+x_{i+k})^{-1}$, where the infimum is taken over all pairs of integers $n\geq k\geq 1$ and all positive $x_1,\dots,x_{n+k}$ subject to cyclicity assumption $x_{n+i}=x_i$, $i=1,\dots,k$. We prove that $\ln 2\leq C< 0.9305$. In the definition of the constant $C$ the operation $\inf_k\inf_n\inf_{\mathbf{x}}$ can be replaced by $\lim_{k\to\infty}\lim_{n\to\infty}\inf_{\mathbf{x}}$.Comment: 12p

The boundary integral equation code PCGrate-S(X) is used to analyze diffraction on Hubble Space Telescope Cosmic Origins Spectrograph gratings at different boundary shapes and layer thicknesses. An effect of resonance anomalies excited in nonconformal dielectric layers overcoated on the surface of metallic grating on the efficiency is studied for the first time to our knowledge. Refractive indices (RIs) for bulk MgF2 taken from well-known references are found to be not suitable for thin optical layers at wavelengths between 115 and 170 nm. A method based on scale fitting of calculated and measured grating efficiencies is outlined for derivation of thin-film optical constants at hard to measure wavelengths. The calculated efficiency based on real boundary profiles and derived RIs of the G185M subwavelength grating is shown to fit within 9.6% or better to the measured data.