The 𝐿𝑝 restriction bounds for Neumann data on surface

Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:msub> <m:mi>u</m:mi> <m:mi>λ</m:mi> </m:msub> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:math> \{uλ\} be a sequence of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> </m:math> L² -normalized Laplacian eigenfunctions on a compact two-dimensional smooth Riemanniann manifold <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>M</m:mi> <m:mo>,</m:mo> <m:mi>g</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> (M,g) . We seek to get an <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>L</m:mi> <m:mi>p</m:mi> </m:msup> </m:math> Lᵖ restriction bound of the Neumann data <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mrow> <m:msup> <m:mi>λ</m:mi> <m:mrow> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:mo lspace="0em">⁢</m:mo> <m:mrow> <m:msub> <m:mo rspace="0em">∂</m:mo> <m:mi>ν</m:mi> </m:msub> <m:msub> <m:mi>u</m:mi> <m:mi>λ</m:mi> </m:msub> </m:mrow> </m:mrow> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mi>γ</m:mi> </m:msub> </m:math> λ⁻¹∂νuλ|γ along a unit geodesic 𝛾. Using the 𝑇- <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>T</m:mi> <m:mo>∗</m:mo> </m:msup> </m:math> T* argument, one can transfer the problem to an estimate of the norm of a Fourier integral operator and show that such bound is <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>O</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi>λ</m:mi> <m:mrow> <m:mrow> <m:mo>−</m:mo> <m:mfrac> <m:mn>1</m:mn> <m:mi>p</m:mi> </m:mfrac> </m:mrow> <m:mo>+</m:mo> <m:mfrac> <m:mn>3</m:mn> <m:mn>2</m:mn> </m:mfrac> </m:mrow> </m:msup> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> O(λ-1/p+3/2) . The stationary phase theorem plays the crucial role in our proof. Moreover, this upper bound is shown to be optimal.